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Prime and Composite Numbers

Key Concepts

Solved Examples: 75

Practice Questions: 7

Solved Examples

Question 1: Determine whether 29 is a prime number.

A prime number has exactly two factors: 1 and the number itself. To check whether 29 is prime, we test divisibility by prime numbers up to √29.
√29 ≈ 5.38
So, we check divisibility by 2, 3, and 5.
29 is not divisible by 2 because it is odd.
29 ÷ 3 leaves a remainder.
29 is not divisible by 5 because it does not end in 0 or 5.
Therefore, 29 has no factors other than 1 and 29. So, 29 is a prime number.

Question 2: Determine whether 51 is a prime number.

To check whether 51 is prime, test divisibility by prime numbers up to √51.
√51 ≈ 7.14
We check divisibility by 2, 3, 5, and 7.
51 is not divisible by 2.
51 ÷ 3 = 17
Therefore, 51 has factors:
1, 3, 17, and 51
Since it has more than two factors, it is not a prime number.

Question 3: Which of the following numbers are prime numbers? 23, 35, 41, 57, 61

Check each number:
23:
○ Not divisible by 2, 3, or 5.
○ Prime number.
35:
○ 35 = 5 × 7
○ Composite number.
41:
○ Not divisible by 2, 3, or 5.
○ Prime number.
57:
○ 57 = 3 × 19
○ Composite number.
61:
○ Not divisible by 2, 3, 5, or 7.
○ Prime number.
The prime numbers are 23, 41, and 61.

Question 4: Determine whether 21 is a composite number.

A composite number has more than two factors.
To check whether 21 is composite, find its factors:
1 × 21 = 21
3 × 7 = 21
Factors of 21 are:
1, 3, 7, and 21
Since 21 has more than two factors, it is a composite number.
21 is a composite number.

Question 5: A shopkeeper buys an article for ¤1,500 and sells it at a profit of 20%. Find the selling price.

Check each number:
14:
○ 14 = 2 × 7
○ Composite number.
17:
○ Has only factors 1 and 17.
○ Prime number.
23:
○ Has only factors 1 and 23.
○ Prime number.
27:
○ 27 = 3 × 9
○ Composite number.
31:
○ Has only factors 1 and 31.
○ Prime number.
39:
○ 39 = 3 × 13
○ Composite number.
The composite numbers are 14, 27, and 39.

Question 6: Find the smallest composite number greater than 10.

The numbers greater than 10 are: 11, 12, 13, 14, …
Check each number:
○ 11 has only two factors (1 and 11), so it is prime.
○ 12 has factors other than 1 and itself:
1, 2, 3, 4, 6, 12
Therefore, 12 is composite.
The smallest composite number greater than 10 is 12.

Question 7: Find the largest composite number less than 20.

Starting from 19 and moving backwards:
○ 19 is prime.
○ 18 has factors other than 1 and itself:
18 = 2 × 9
Therefore, 18 is composite.
So, the largest composite number less than 20 is 18.

Question 8: How many composite numbers are there between 1 and 20?

Numbers from 1 to 20:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Prime numbers:
2, 3, 5, 7, 11, 13, 17, 19
The number 1 is neither prime nor composite.
Composite numbers:
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20
There are 11 composite numbers between 1 and 20.

Question 9: Which is the smallest composite number that is odd?

The smallest composite numbers are:
4, 6, 8, 9, …
Among these, 9 is the first odd number.
Factors of 9: 1, 3, 9
Since 9 has more than two factors, it is composite.
The smallest odd composite number is 9.

Question 10: Find all prime numbers between 10 and 30.

The numbers between 10 and 30 are:
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29
Checking the numbers:
○ 11 → Prime
○ 13 → Prime
○ 17 → Prime
○ 19 → Prime
○ 23 → Prime
○ 29 → Prime
The remaining numbers have factors other than 1 and themselves.
The prime numbers between 10 and 30 are: 11, 13, 17, 19, 23, and 29

Question 11: How many prime numbers are there between 1 and 100?

he prime numbers between 1 and 100 are:
2, 3, 5, 7, 11, 13, 17, 19, 23,
29, 31, 37, 41, 43, 47, 53, 59,
61, 67, 71, 73, 79, 83, 89, 97
Total prime numbers = 25

Question 12: Find the smallest prime number greater than 10.

Prime numbers greater than 10 are checked in order: 11, 12, 13, …
○ 11 has only two factors: 1 and 11.
○ Therefore, 11 is a prime number.
The smallest prime number greater than 10 is 11.

Question 13: Find the largest prime number less than 50.

Starting from 50 and moving backwards:
○ 50 → Composite
○ 49 = 7 × 7 → Composite
○ 48 → Composite
○ 47 → Prime
Since 47 has only two factors, 1 and 47, it is the largest prime number below 50.
The largest prime number less than 50 is 47.

Question 14: Identify the even prime number.

A prime number has exactly two factors: 1 and the number itself.
The smallest even numbers are: 2, 4, 6, 8, 10, …
Among these, only 2 has exactly two factors:
Factors of 2: 1 and 2
All other even numbers are divisible by 2 and have more than two factors.
The only even prime number is 2.

Question 15: Why is 2 the only even prime number?

Every even number greater than 2 is divisible by 2.
For example:
○ 6 = 2 × 3
○ 10 = 2 × 5
○ 14 = 2 × 7
Therefore, every even number greater than 2 has at least three factors:
1, 2, and the number itself.
A prime number can have only two factors.
2 is the only even prime number because all other even numbers are composite.

Question 16: Find the sum of all odd prime numbers between 1 and 20.

Prime numbers between 1 and 20: 2, 3, 5, 7, 11, 13, 17, 19
Odd prime numbers: 3, 5, 7, 11, 13, 17, 19
Sum: 3 + 5 + 7 + 11 + 13 + 17 + 19 = 75
The sum of odd prime numbers between 1 and 20 is 75.

Question 17: Identify the twin prime pairs between 1 and 30.

Twin prime numbers are pairs of prime numbers that differ by 2.
Prime numbers between 1 and 30 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Check pairs with a difference of 2:
○ 3 − 2 = 1 ❌
○ 5 − 3 = 2 ✓
○ 7 − 5 = 2 ✓
○ 11 − 7 = 4 ❌
○ 13 − 11 = 2 ✓
○ 17 − 13 = 4 ❌
○ 19 − 17 = 2 ✓
○ 23 − 19 = 4 ❌
○ 29 − 23 = 6 ❌
The twin prime pairs between 1 and 30 are: (3, 5), (5, 7), (11, 13), and (17, 19)

Question 18: Identify the twin prime pairs from the following numbers: (11, 13), (19, 21), (29, 31), (41, 43)

Check each pair:
(11, 13):
○ Both are prime.
○ Difference = 13 − 11 = 2 ✓
(19, 21):
○ 21 is composite.
○ Not a twin prime pair. ❌
(29, 31):
○ Both are prime.
○ Difference = 31 − 29 = 2 ✓
(41, 43):
○ Both are prime.
○ Difference = 43 − 41 = 2 ✓
The twin prime pairs are: (11, 13), (29, 31), and (41, 43)

Question 19: Check whether 8 and 15 are co-prime numbers.

Two numbers are co-prime if their only common factor is 1.
Factors of 8: 1, 2, 4, 8
Factors of 15: 1, 3, 5, 15
The only common factor is 1.
8 and 15 are co-prime numbers.

Question 20: Check whether 12 and 18 are co-prime numbers.

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6
Since they have common factors other than 1, they are not co-prime.
12 and 18 are not co-prime numbers.

Question 21: Identify the co-prime pairs from the following: (5, 12), (8, 20), (14, 25), (18, 27)

(5, 12):
○ Common factor = 1 ✓
○ Co-prime
(8, 20):
○ Common factors include 2 and 4 ❌
○ Not co-prime
(14, 25):
○ Common factor = 1 ✓
○ Co-prime
(18, 27):
○ Common factors include 3 and 9 ❌
○ Not co-prime
The co-prime pairs are: (5, 12) and (14, 25)

Question 22: Find the co-prime numbers between 1 and 10 that are co-prime with 10.

Numbers between 1 and 10: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Factors of 10: 1, 2, 5, 10
Numbers having no common factor with 10 except 1 are:
1, 3, 7, 9
The numbers co-prime with 10 between 1 and 10 are: 1, 3, 7, and 9

Question 23: Find the greatest number less than 20 that is co-prime with 20.

Factors of 20: 1, 2, 4, 5, 10, 20
Check numbers below 20 starting from the largest 19:
Factors of 19: 1 and 19
Common factor with 20 = 1
Therefore, 19 is co-prime with 20.
The greatest number less than 20 that is co-prime with 20 is 19.

Question 24: Check whether 35 and 64 are co-prime numbers.

Factors of 35: 1, 5, 7, 35
Factors of 64: 1, 2, 4, 8, 16, 32, 64
The only common factor is 1.
So, 35 and 64 are co-prime numbers.

Question 25: The sum of two prime numbers is 36. Find all such possible pair of prime numbers.

List of prime numbers between 2 and 36: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 33
Next, we subtract each prime from 36 and check if the result is also a prime number:
○ 36 – 2 = 34 ❌
○ 36 – 3 = 33 ❌
○ 36 – 5 = 31 ✓
○ 36 – 7 = 29 ✓
○ 36 – 11 = 25 ❌
○ 36 – 13 = 23 ✓
○ 36 – 17 = 19 ✓
Complete list of valid pairs: (5, 31), (7, 29), (13, 23) and (17, 19)

Question 26: Which of the following statements is always true?
A) Every odd number is prime.
B) Every prime number is odd.
C) Every composite number has more than two factors.
D) Every even number is composite.

A) False. For example, 9 is odd but composite.
B) False. 2 is an even prime number.
C) True. A composite number has more than two factors.
D) False. 2 is even but prime.
Correct option C) Every composite number has more than two factors.

Question 27: Which of the following numbers is neither prime nor composite?
A) 0
B) 1
C) 2
D) 3

A prime number has exactly two factors, while a composite number has more than two factors.
The number 1 has only one factor, itself, so it is neither prime nor composite.
Correct option is B

Question 28: Which of the following pairs consists of two prime numbers?
A) (21, 23)
B) (29, 31)
C) (35, 37)
D) (39, 41)

A) 21 is composite
B) 29 and 31 are both prime
C) 35 is composite
D) 39 is composite
So the correct option is B

Question 29: Which of the following statements is correct?
A) Every prime number is a co-prime number.
B) Every pair of co-prime numbers is prime.
C) Two composite numbers can be co-prime.
D) Every pair of prime numbers is a twin prime pair.

A) False. A prime number is a single number, whereas co-prime refers to a pair of numbers. These are different concepts.
B) False. Co-prime numbers need not be prime. For example, 8 and 15 are co-prime, but both are not prime numbers.
C) True. Two composite numbers can be co-prime if their only common factor is 1. For example, 8 and 9 are both composite and are co-prime.
D) False. Every pair of prime numbers is not a twin prime pair. For example, 13 and 19 are both prime, but their difference is 6, not 2.
The correct option is C) Two composite numbers can be co-prime.

Question 30: Which one of the following is a twin prime pair?
A) (13, 15)
B) (17, 19)
C) (29, 33)
D) (41, 45)

A) False. Although the difference is 2, 15 is a composite number.
B) True. Both 17 and 19 are prime numbers, and their difference is 2.
C) False. 33 is divisible by 3 and 11, so it is not prime.
D) False. 45 is divisible by 3 and 5, and the difference between 41 and 45 is 4.
Correct option is B) (17, 19)

Question 31: Which of the following statements is false?
A) Every prime number greater than 2 is odd.
B) The smallest prime number is 2.
C) Every odd number greater than 2 is prime.
D) 2 is the only even prime number.

A) True. Every prime number greater than 2 is odd.
B) True. The smallest prime number is 2.
C) False. Numbers such as 9, 15, and 21 are odd but composite.
D) True. Every even number greater than 2 is divisible by 2 and is therefore composite.
Correct option is C) Every odd number greater than 2 is prime.

Question 32: Which of the following pairs is not co-prime?
A) (7, 10)
B) (8, 15)
C) (18, 27)
D) (11, 12)

A) False. The only common factor of 7 and 10 is 1, so they are co-prime.
B) False. The only common factor of 8 and 15 is 1, so they are co-prime.
C) True. 18 and 27 have common factors 1, 3, and 9, so they are not co-prime.
D) False. The only common factor of 11 and 12 is 1, so they are co-prime.
Correct option is C) (18, 27)

Question 33: If a number has exactly two distinct positive factors, then the number is:
A) Composite
B) Prime
C) Co-prime
D) Twin prime

A) False. A composite number has more than two positive factors.
B) True. A prime number has exactly two distinct positive factors: 1 and the number itself.
C) False. Co-prime refers to a pair of numbers whose only common factor is 1, not a single number.
D) False. Twin prime refers to a pair of prime numbers whose difference is 2.
Correct option is B

Question 34: Which of the following statements is always true?
A) Every pair of consecutive numbers is co-prime.
B) Every pair of consecutive numbers is prime.
C) Every pair of consecutive numbers is composite.
D) Every pair of consecutive numbers is a twin prime pair.

A) True. Consecutive numbers differ by 1, so they cannot have any common factor greater than 1. Therefore, they are always co-prime.
B) False. For example, 8 and 9 are consecutive numbers, but neither is prime.
C) False. For example, 2 and 3 are consecutive numbers, and both are prime.
D) False. Twin prime numbers differ by 2, whereas consecutive numbers differ by 1.
So correct option is A) Every pair of consecutive numbers is co-prime.

Question 35: Which of the following statements is correct?
A) Every prime number has an odd number of factors.
B) Every composite number has exactly four factors.
C) Every prime number has exactly two factors.
D) Every composite number is divisible only by 1 and itself.

A) False. Every prime number has exactly two factors.
B) False. Composite numbers may have three, four, or more factors. For example, 12 has six factors.
C) True. A prime number has exactly two factors: 1 and itself.
D) False. A composite number is divisible by at least one number other than 1 and itself.
So the correct option is C) Every prime number has exactly two factors.

Question 36: Which of the following pairs cannot be co-prime?
A) Two consecutive numbers
B) Two even numbers
C) An even number and an odd number
D) A prime number and a composite number

A) False. Consecutive numbers are always co-prime.
B) True. Any two even numbers have at least 2 as a common factor, so they cannot be co-prime.
C) False. For example, 8 and 15 are co-prime.
D) False. For example, 5 and 6 are co-prime.
The correct option is B) Two even numbers

Question 37: Which of the following statements is false?
A) Every prime number except 2 is odd.
B) The number 1 is neither prime nor composite.
C) Every composite number has at least one prime factor.
D) Every odd number is prime.

A) True. All prime numbers greater than 2 are odd.
B) True. The number 1 has only one factor, so it is neither prime nor composite.
C) True. Every composite number can be expressed as a product of prime factors.
D) False. Numbers such as 9, 15, and 21 are odd but composite.
Correct option is D) Every odd number is prime.

Question 38: Which of the following statements is correct?
A) The smallest composite number is 2.
B) The smallest prime number is 1.
C) The smallest composite number is 4.
D) The smallest odd prime number is 5.

A) False. 2 is the smallest prime number.
B) False. 1 is neither prime nor composite.
C) True. 4 has factors 1, 2, and 4, making it the smallest composite number.
D) False. The smallest odd prime number is 3.
The correct option is C) The smallest composite number is 4.

Question 39: Which of the following statements is always true?
A) Every prime number is co-prime with every other prime number.
B) Every pair of composite numbers is co-prime.
C) Every pair of odd numbers is co-prime.
D) Every pair of even numbers is co-prime.

A) True. Any two distinct prime numbers have no common factor other than 1. Therefore, they are co-prime.
B) False. For example, 8 and 12 have common factors 2 and 4.
C) False. For example, 9 and 15 have a common factor of 3.
D) False. Every pair of even numbers has at least 2 as a common factor.
Correct option is A) Every prime number is co-prime with every other prime number.

Question 40: Which of the following statements is false?
A) A prime number can also be a co-prime number with another number.
B) A composite number can be co-prime with another composite number.
C) Every pair of twin prime numbers is co-prime.
D) Every pair of co-prime numbers is a twin prime pair.

A) True. For example, 5 and 6 are co-prime.
B) True. For example, 8 and 9 are both composite and co-prime.
C) True. Twin prime pairs such as 17 and 19 have no common factor other than 1.
D) False. Numbers such as 8 and 15 are co-prime but are not twin prime numbers.
So the correct option is D) Every pair of co-prime numbers is a twin prime pair.

Question 41: Which of the following statements is correct?
A) Every composite number is divisible by at least one prime number.
B) Every prime number is divisible by 2.
C) Every odd number is prime.
D) Every co-prime pair consists of two prime numbers.

A) True. Every composite number has at least one prime factor. For example, 18 is divisible by the prime numbers 2 and 3.
B) False. Only the prime number 2 is divisible by 2. Prime numbers such as 3, 5, and 7 are not divisible by 2.
C) False. Numbers such as 9 and 15 are odd but composite.
D) False. Numbers such as 8 and 15 are co-prime, although neither is a prime number.
The correct option is A) Every composite number is divisible by at least one prime number.

Question 42: Which of the following statements is correct?
A) Every prime number has exactly one factor.
B) Every composite number has exactly two factors.
C) Every prime number has exactly two factors.
D) Every composite number is divisible only by 1 and itself.

A) False. Every prime number has exactly two factors: 1 and itself.
B) False. Composite numbers have more than two factors. For example, 12 has six factors.
C) True. A prime number always has exactly two factors: 1 and itself.
D) False. A composite number is divisible by at least one number other than 1 and itself.
The correct option is C) Every prime number has exactly two factors.

Question 44: Which of the following statements is correct?
A) Every multiple of a prime number is prime.
B) Every multiple of a prime number greater than the prime itself is composite.
C) Every multiple of 2 is prime.
D) Every multiple of 3 is prime.

A) False. For example, 15 is a multiple of 5 but is composite.
B) True. Every multiple of a prime number greater than the prime itself has at least three factors. For example, 21 is a multiple of 7 and is composite.
C) False. Numbers such as 4, 6, and 8 are multiples of 2 but are composite.
D) False. Numbers such as 9 and 15 are multiples of 3 but are composite.
The correct option is B) Every multiple of a prime number greater than the prime itself is composite.

Question 44: If p and p + 2 are twin primes and both are greater than 3, what must always be true about the number right between them, p + 1?

Since p and p + 2 are twin primes greater than 3, both numbers must be odd.
The number between them is: p + 1
○ The sum of an odd number and 1 is always even. Therefore, p + 1 is divisible by 2.
○ Also, among any three consecutive numbers: p, p + 1, p + 2 and one number must be divisible by 3.
○ Since p and p + 2 are prime numbers greater than 3, neither can be divisible by 3. Therefore, p + 1 must be divisible by 3.
Hence, p + 1 is divisible by both 2 and 3.

Question 45: A prime number is greater than 10. When its digits are added together, the sum is a multiple of 11. What is the smallest possible prime number that satisfies this condition?

We need to find the smallest prime number greater than 10 whose digit sum is a multiple of 11.
Check prime numbers starting from the smallest:
○ 11 → Digit sum = 1 + 1 = 2 (Not a multiple of 11)
○ 13 → Digit sum = 1 + 3 = 4 (Not a multiple of 11)
○ 17 → Digit sum = 1 + 7 = 8 (Not a multiple of 11)
○ 19 → Digit sum = 1 + 9 = 10 (Not a multiple of 11)
○ 23 → Digit sum = 2 + 3 = 5 (Not a multiple of 11)
○ 29 → Digit sum = 2 + 9 = 11 (Multiple of 11)
Since 29 is a prime number and its digit sum is 11, it satisfies the given condition.
So the smallest possible prime number is 29.

Question 46: Express 100 as the sum of odd prime numbers. [Note (3, 97) and (97, 3) are to be considered the same pair.]

Let’s test odd primes and subtract them from 100 to see if the leftover is prime:
○ 100 − 3 = 97 (prime)
○ 100 − 5 = 95 (not a prime)
○ 100 − 7 = 93 (not a prime)
○ 100 − 11 = 89 (prime)
○ 100 − 13 = 87 (not a prime)
○ 100 − 17 = 83 (prime)
○ 100 − 19 = 81 (not a prime)
○ 100 − 23 = 77 (not a prime)
○ 100 − 29 = 71 (prime)
○ 100 − 31 = 69 (not a prime)
○ 100 − 37 = 63 (not a prime)
○ 100 − 41 = 59 (prime)
○ 100 − 43 = 57 (not a prime)
○ 100 − 47 = 53 (prime)
So the pairs are (3, 97), (11, 89), (17, 83), (29, 71), (41, 59), (47, 53)

Question 48: What is the smallest two-digit prime number that remains a prime number when you reverse its digits, but becomes a composite number if you add 2 to it?

Two-digit primes that are still prime when reversed (reversible primes): 11, 13, 17, 31, 37, 71, 73, 79, 97.
Adding 2 to each starting from 11
○ 11 + 2 = 13 (Prime)
○ 13 + 5 = 15 (Composite)
So 13 is the smallest 2 digit prime number that remains a prime when reversed but becomes a composite when 2 is added to it.

Question 48: What is the smallest positive integer that has exactly 3 positive divisors (factors)?

1 → Factors: 1 (only 1 factor)
2 → Factors: 1, 2 (2 factors)
3 → Factors: 1, 3 (2 factors)
4 → Factors: 1, 2, 4 (3 factors)
The smallest positive integer with exactly 3 positive divisors is 4.

Question 49: If two twin primes are both greater than 10, what is the only digit that their product can never end with?

Any prime number greater than 5 can only end with: 1, 3, 7, or 9
For twin primes, the difference between the two primes is 2.
Possible ending digit pairs are:
(1, 3)
○ Example: 11 and 13
○ Product ending: 1 × 3 = 3
(7, 9)
○ Example: 17 and 19
○ Product ending: 7 × 9 = 63 → ends in 3
(9, 1)
○ Example: 29 and 31
○ Product ending: 9 × 1 = 9
The pair (3, 5) and (5, 7) are impossible because a prime greater than 5 cannot end in 5.
Therefore, the product of two twin primes greater than 10 can only end in: 3 or 9
Hence the product of two twin primes greater than 10 can never end with the digits 1, 5, or 7.

Question 50: What is the smallest composite number that does not have any prime factors less than 10?

Prime numbers less than 10 are: 2, 3, 5, and 7
The required composite number should not be divisible by 2, 3, 5, or 7.
A composite number must have prime factors. Therefore, its prime factors must be 10 or greater.
The smallest prime number greater than 10 is: 11
The smallest composite number using only this prime factor is: 11 × 11 = 121
It is composite and its only prime factor is 11, which is not less than 10.

Question 51: The area of a rectangle is 31 square centimeters. If the lengths of the sides are whole numbers, what is the perimeter of the rectangle?

Area = Length × Breadth
Given: Length × Breadth = 31
Since 31 is a prime number, its only positive factors are:
1 and 31
Therefore, the possible side lengths of the rectangle are:
Length = 31 cm
Breadth = 1 cm
The perimeter of a rectangle is:
Perimeter = 2 × (Length + Breadth)
= 2 × (31 + 1)
= 2 × 32
= 64 cm
So the perimeter of the rectangle is 64 cm.

Question 52: How many two-digit composite numbers have a square root that is a prime number?

A number whose square root is a prime number must be the square of a prime number.
The two-digit numbers lie between: 10 and 99
We need to find prime numbers whose squares fall in this range.
The prime numbers are:
2, 3, 5, 7, 11, …
Now check their squares:
2² = 4 → Not a two-digit number
3² = 9 → Not a two-digit number
5² = 25 → Two-digit composite number ✓
7² = 49 → Two-digit composite number ✓
11² = 121 → Three-digit number
Therefore, the two-digit composite numbers with prime square roots are:
25 and 49
Number of such numbers: 2

Question 53: A composite number N has exactly 4 factors. If N is multiplied by a prime number p, what is the minimum possible number of factors of the new number (N × p)?

A number with exactly 4 factors can have the following prime factor forms:
N = a³
or
N = a × b
where a and b are prime numbers.
To get the minimum possible number of factors after multiplying by a prime number, we should choose a prime number that is already a factor of N.
Example:
Let: N = 2³ = 8
Factors of 8: 1, 2, 4, 8
So, N has exactly 4 factors.
Now multiply by the same prime number:
N × p = 8 × 2 = 16
Prime factor form:
16 = 2⁴
Number of factors:
= 4 + 1
= 5
Factors of 16: 1, 2, 4, 8, 16
Therefore, the minimum possible number of factors is: 5

Question 54: Find a whole number n between 1 and 10 that makes the expression n² + n + 1 a composite number.

We need to find a value of n between 1 and 10 such that: n² + n + 1 is composite.
For n = 1:
1² + 1 + 1 = 3 (3 is prime) ❌
For n = 2:
2² + 2 + 1 = 4 + 2 + 1 = 7 (7 is prime) ❌
For n = 3:
3² + 3 + 1 = 9 + 3 + 1 = 13 (13 is prime) ❌
For n = 4:
4² + 4 + 1 = 16 + 4 + 1 = 21 (21 = 3 × 7)
Therefore, 21 is composite. ✓
Hence, n = 4 satisfies the condition.

Question 55: If the product of three different prime numbers is equal to 5 times their sum, what is the largest of these three prime numbers?

Let the three prime numbers be a, b, and c.
Given: a × b × c = 5 × (a + b + c)
Since the right side contains the prime factor 5, one of the three primes must be 5.
Let: a = 5
Then:
5 × b × c = 5 × (5 + b + c)
Dividing by 5:
b × c = 5 + b + c
Rearranging:
bc − b − c = 5
Adding 1 to both sides:
(b − 1)(c − 1) = 6
The factor pairs of 6 are:
1 × 6 and 2 × 3
Checking the possibilities:
b − 1 = 1, c − 1 = 6
⇒ b = 2, c = 7 ✓
b − 1 = 2, c − 1 = 3
⇒ b = 3, c = 4 (not prime) ✗
Therefore, the three prime numbers are: 2, 5, and 7
Largest prime number = 7

Question 56: Two positive integers x and y are coprime. If their Least Common Multiple (LCM) is 105, what is the smallest possible value for their sum x + y?

The prime factorisation of 105 is: 105 = 3 × 5 × 7
Since x and y are co-prime, they cannot share any common prime factor. Therefore, the prime factors of 105 must be divided between x and y.
To get the smallest possible sum, we should divide the factors as evenly as possible.
Possible pairs are:
(3, 35)
○ Sum = 3 + 35 = 38
(5, 21)
○ Sum = 5 + 21 = 26
(7, 15)
○ Sum = 7 + 15 = 22
x = 7 and y = 15
Check:
HCF(7, 15) = 1 ✓
LCM(7, 15) = 7 × 15 = 105 ✓
Therefore: x + y = 7 + 15 = 22
So the smallest possible value of x + y is 22.

Question 57: If n is a positive integer such that n + 1 and n + 3 are both prime numbers, how many possible values can n take?

We need two numbers that are 2 apart and both prime.
That means: n + 1 and n + 3 must be a pair of twin primes.
Examples
If:
n + 1 = 3 and n + 3 = 5
Then:
n = 2
If:
n + 1 = 5 and n + 3 = 7
Then:
n = 4
If:
n + 1 = 11 and n + 3 = 13
Then:
n = 10
If:
n + 1 = 17 and n + 3 = 19
Then:
n = 16
There are many such pairs:
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), …
Therefore, there are infinitely many possible values of n if twin primes are infinite.

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Question 58: The number 143 is a composite number formed by multiplying two prime numbers p and q. What is the value of the expression: (p − 1) × (q − 1)?

First, express 143 as a product of two prime numbers: 143 = 11 × 13
Therefore:
p = 11 and q = 13
Now substitute the values:
(p − 1) × (q − 1)
= (11 − 1) × (13 − 1)
= 10 × 12
= 120

Question 59: A page number in a book is a two-digit composite number. When the digits of the page number are multiplied together, the result is a prime number. How many page numbers satisfy this condition?

Let the two-digit number be AB.
The product of the digits is: A × B
For A × B to be a prime number, one digit must be 1 and the other must be a prime digit:
2, 3, 5, or 7
Possible numbers: 12, 13, 15, 17, 21, 31, 51, 71
Now check which of these are composite:
12 = 3 × 4 → Composite ✓
13 → Prime ✗
15 = 3 × 5 → Composite ✓
17 → Prime ✗
21 = 3 × 7 → Composite ✓
31 → Prime ✗
51 = 3 × 17 → Composite ✓
71 → Prime ✗
Therefore, the valid page numbers are: 12, 15, 21, and 51
So the number of pages that satisfy the condition are 4

Question 60:Max picks two twin prime numbers that are both less than 10. When he adds them together, the sum is a multiple of 4. Which two numbers did Max pick?

The prime numbers less than 10 are: 2, 3, 5, 7
Twin prime pairs are pairs of prime numbers that differ by 2.
The twin prime pairs less than 10 are:
(3, 5)
(5, 7)
Now check their sums:
For 3 and 5:
3 + 5 = 8
8 is a multiple of 4. ✓
For 5 and 7:
5 + 7 = 12
12 is also a multiple of 4. ✓
Therefore, both pairs satisfy the condition.
So the possible pairs are (3, 5) and (5, 7).

Question 61: Can the sum of two different prime numbers ever be a prime number itself?

Consider two different prime numbers. Except for 2, all prime numbers are odd.
When two odd prime numbers are added the result is even.
For example:
3 + 5 = 8
5 + 7 = 12
The result is an even number greater than 2, so it cannot be prime.
Now consider the only even prime number: 2
Adding 2 to an odd prime gives: 2 + 3 = 5
5 is a prime number.
Therefore, the sum of two different prime numbers can be a prime number, but only when one of the primes is 2.

Question 62: Three distinct prime numbers a, b, and c satisfy the equation: abc = 7(a + b + c)
Find the largest of the three prime numbers.

Since the right side is divisible by 7, one of the primes must be 7.
Let a = 7.
Then: 7bc = 7(7 + b + c)
Dividing by 7:
bc = 7 + b + c
Rearranging:
bc − b − c = 7
Adding 1 to both sides:
(b − 1)(c − 1) = 8
Factor pairs of 8:
1 × 8 ⇒ b = 2, c = 9 (9 is not prime) ✗
2 × 4 ⇒ b = 3, c = 5 ✓
Thus, the three primes are: 3, 5, and 7
So the largest of 3 primes is 7

Question 63: Let p and q be twin prime numbers such that 5 < p < q < 30. Find the sum of the products p × q for all possible pairs of (p, q).[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row admin_label="SE_Answer" _builder_version="4.27.6" _module_preset="default" custom_margin="-50px||||false|false" global_colors_info="{}"][et_pb_column type="4_4" _builder_version="4.27.6" _module_preset="default" global_colors_info="{}"][et_pb_text _builder_version="4.27.6" _module_preset="default" text_font="|300|||||||" text_font_size="18px" custom_padding="0.5%||0.5%|1%|false|false" text_font_size_tablet="17px" text_font_size_phone="16px" text_font_size_last_edited="on|phone" border_radii="on|3px|3px|3px|3px" border_width_all="1px" border_color_all="#d97706" border_width_all_tablet="1px" border_width_all_phone="2px" border_width_all_last_edited="on|phone" global_colors_info="{}"]The twin prime pairs satisfying 5 < p < q < 30 are: (11, 13) (17, 19) (29, 31) ✗ since 31 > 30
Their products are:
11 × 13 = 143
17 × 19 = 323
Sum of the products: 143 + 323 = 466

Question 64: Find the number of positive integers n ≤ 50 for which the expression n² + 3n + 2 is a composite number that is co-prime to 6.

n² + 3n + 2 = (n + 1)(n + 2)
Since n + 1 and n + 2 are consecutive integers, one of them is always even.
Therefore, (n + 1)(n + 2) is always even, so it is always divisible by 2.
Also, for every positive integer n, both n + 1 and n + 2 are greater than 1. Hence, their product is always composite.
A number that is co-prime to 6 must not be divisible by 2 or 3.
Since (n + 1)(n + 2) is always divisible by 2, it can never be co-prime to 6.
Therefore, there is no positive integer n ≤ 50 for which n² + 3n + 2 is both composite and co-prime to 6.

Question 65: How many twin prime pairs (p, q) exist such that both prime numbers are strictly between 50 and 80?

List all the odd numbers strictly between 50 and 80:
51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79
Now keep only the prime numbers:
53, 59, 61, 67, 71, 73, 79
Twin prime pairs differ by 2.
The twin prime pairs are:
(59, 61)
(71, 73)
Therefore, there are 2 twin prime pairs.

Question 66: Find the sum of all positive integers n such that n ≠ 3, and the expression (n² − 9)/(n − 3) evaluates to a prime number less than 20.

Factor the numerator:
n² − 9 = (n − 3)(n + 3)
Since n ≠ 3, we can simplify the expression:
(n² − 9)/(n − 3) = n + 3
The expression must be a prime number less than 20.
The prime numbers less than 20 are:
2, 3, 5, 7, 11, 13, 17, 19
Since n is a positive integer, n + 3 > 3. Therefore, the possible values of n + 3 are:
5, 7, 11, 13, 17, 19
Hence,
n = 2
n = 4
n = 8
n = 10
n = 14
n = 16
Sum of all possible values of n:
2 + 4 + 8 + 10 + 14 + 16 = 54

Question 67: Find the number of pairs of prime numbers (p, q) that satisfy the equation: 3p + 4q = 41

Since 4q is even and 41 is odd, 3p must be odd.
As 3 is odd, p must also be odd.
The odd prime numbers less than 41 are: 3, 5, 7, 11, 13
Substitute each value of p into the equation:
p = 3:
○ 3 × 3 + 4q = 41
○ 9 + 4q = 41
○ 4q = 32
○ q = 8 (not prime) ✗
p = 5:
○ 3 × 5 + 4q = 41
○ 15 + 4q = 41
○ 4q = 26
○ q = 6.5 (not an integer) ✗
p = 7:
○ 3 × 7 + 4q = 41
○ 21 + 4q = 41
○ 4q = 20
○ q = 5 (prime) ✓
p = 11:
○ 3 × 11 + 4q = 41
○ 33 + 4q = 41
○ 4q = 8
○ q = 2 (prime) ✓
p = 13:
○ 3 × 13 + 4q = 41
○ 39 + 4q = 41
○ 4q = 2
○ q = 0.5 (not an integer) ✗
Therefore, the prime pairs are: (7, 5) and (11, 2)
Hence, the number of such pairs is: 2

Question 68: A token starts at position 5 on a coordinate line. In each step, if it is at position x, it can jump to either: 2x + 1 or 2x + 2. What is the minimum number of steps required to reach a position that is a composite number co-prime to 3?

Starting position: x = 5
Step 1:
Possible positions:
2(5) + 1 = 11
2(5) + 2 = 12
○ 11 is prime. ✗
○ 12 is composite, but it is divisible by 3. ✗
Step 2:
From 11:
2(11) + 1 = 23
2(11) + 2 = 24
○ 23 is prime. ✗
○ 24 is composite, but divisible by 3. ✗
From 12:
2(12) + 1 = 25
2(12) + 2 = 26
○ 25 is composite and:
○ HCF(25, 3) = 1 ✓
Therefore, 25 is co-prime to 3.
The token reaches the required position in 2 steps.

Question 69: What is the value of the largest prime number p such that all integers strictly between p and p + 6 are composite, given that p < 30?[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row admin_label="SE_Answer" _builder_version="4.27.6" _module_preset="default" custom_margin="-50px||||false|false" global_colors_info="{}"][et_pb_column type="4_4" _builder_version="4.27.6" _module_preset="default" global_colors_info="{}"][et_pb_text _builder_version="4.27.6" _module_preset="default" text_font="|300|||||||" text_font_size="18px" custom_padding="0.5%||0.5%|1%|false|false" text_font_size_tablet="17px" text_font_size_phone="16px" text_font_size_last_edited="on|phone" border_radii="on|3px|3px|3px|3px" border_width_all="1px" border_color_all="#d97706" border_width_all_tablet="1px" border_width_all_phone="2px" border_width_all_last_edited="on|phone" global_colors_info="{}"]We need to find a prime number p where the numbers: p + 1, p + 2, p + 3, p + 4, p + 5 are all composite. The prime numbers less than 30 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 Check from the largest prime downward: For p = 29: Numbers between 29 and 35: 30, 31, 32, 33, 34 But 31 is prime. ✗ For p = 23: Numbers between 23 and 29: 24, 25, 26, 27, 28 Check: 24 = 2 × 12 (Composite) 25 = 5 × 5 (Composite) 26 = 2 × 13 (Composite) 27 = 3 × 9 (Composite) 28 = 4 × 7 (Composite) All five numbers are composite. ✓ Therefore, the largest possible value of p is 23[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row admin_label="SE_Question" _builder_version="4.27.6" _module_preset="default" custom_margin="-50px||||false|false" global_colors_info="{}"][et_pb_column type="4_4" _builder_version="4.27.6" _module_preset="default" global_colors_info="{}"][et_pb_text _builder_version="4.27.6" _module_preset="default" text_font="|300|||||||" text_font_size="18px" custom_padding="0.5%||0.5%|1%|false|false" text_font_size_tablet="17px" text_font_size_phone="16px" text_font_size_last_edited="on|phone" border_radii="on|3px|3px|3px|3px" border_width_all="1px" border_color_all="#0d9488" border_width_all_tablet="1px" border_width_all_phone="2px" border_width_all_last_edited="on|phone" global_colors_info="{}"]Question 70: For how many positive integers n is the expression 6n ÷ (n + 2) a prime number?

Rewrite the expression: 6n ÷ (n + 2)
Since
6n = 6(n + 2) − 12
we get:
6n ÷ (n + 2) = [6(n + 2) − 12] ÷ (n + 2)
= 6 − 12 ÷ (n + 2)
For the expression to be a prime number, 12 ÷ (n + 2) must be an integer.
Therefore, n + 2 must be a divisor of 12.
Positive divisors of 12 greater than or equal to 3 are 3, 4, 6, 12
Checking these values:
For n + 2 = 3:
n = 1
Expression = 6 − 12 ÷ 3 = 6 − 4 = 2 (prime) ✓
For n + 2 = 4:
n = 2
Expression = 6 − 12 ÷ 4 = 6 − 3 = 3 (prime) ✓
For n + 2 = 6:
n = 4
Expression = 6 − 12 ÷ 6 = 6 − 2 = 4 (not prime) ✗
For n + 2 = 12:
n = 10
Expression = 6 − 12 ÷ 12 = 6 − 1 = 5 (prime) ✓
Therefore, the possible values of n are 1, 2, and 10
Hence, the number of positive integers n is 3

Question 71: The area of a right-angled triangle is 30 square units. If the lengths of all three sides are integers, how many of the three side lengths are composite numbers?

Let the two perpendicular sides of the right-angled triangle be a and b.
Area of a right-angled triangle: (1/2) × a × b = 30
Therefore a × b = 60
The integer factor pairs of 60 that can form a right-angled triangle are (6, 10)
Now find the hypotenuse using the Pythagorean theorem:
c² = 6² + 10²
c² = 36 + 100
c² = 136
Since 136 is not a perfect square, (6, 10) does not form a right-angled triangle.
Try another factor pair (5, 12)
Hypotenuse c² = 5² + 12²
c² = 25 + 144
c² = 169
c = 13
Therefore, the three side lengths are 5, 12, and 13
Now classify them:
5 is prime.
12 is composite.
13 is prime.
Only one side length is composite.

Question 72: Two twin primes have a sum of 84. Find the two prime numbers.

Twin primes differ by 2.
Let the smaller prime be p.
The larger prime is p + 2
Their sum is p + (p + 2) = 84
2p + 2 = 84
2p = 82
p = 41
The two numbers are 41 and 43
Both are prime and differ by 2.

Question 73: A number leaves remainder 1 when divided by 2, 3, and 5. If the number is a prime number less than 50, find all possible values.

A number leaving remainder 1 when divided by 2, 3, and 5 means the number is of the form:
LCM(2, 3, 5) × k + 1
LCM of 2, 3, and 5 is 2 × 3 × 5 = 30
Therefore:
Number = 30k + 1
Prime numbers less than 50 of this form are:
For k = 0:
Number = 1 (not prime)
For k = 1:
Number = 31 (prime)
For k = 2:
Number = 61 (greater than 50)
Therefore, the only possible value is 31

Question 74: The sum of two twin prime numbers is multiplied by their difference. If the final result is 48, what is the value of the larger prime number?

Let the twin prime numbers be p and p + 2
Their difference is (p + 2) − p = 2
Their sum is:
p + (p + 2) = 2p + 2
Given (Sum) × (Difference) = 48
Therefore
(2p + 2) × 2 = 48
4p + 4 = 48
4p = 44
p = 11
The two twin primes are 11 and 13
The larger prime number is 13

Question 75: The ages of two friends are both composite numbers. The difference between their ages is equal to the product of a pair of twin primes that are both less than 10. If the younger friend is 12 years old, what is the youngest possible age of the older friend?

The twin prime pairs less than 10 are (3, 5) and (5, 7)
Their products are
3 × 5 = 15
5 × 7 = 35
Possible age differences 15 and 35
For a difference of 15:
Older friend’s age is 12 + 15 = 27
27 = 3 × 9, so it is composite. ✓
For a difference of 35:
Older friend’s age 12 + 35 = 47
47 is prime. ✗
Therefore, the youngest possible age of the older friend is 27 years

What are Prime and Composite Numbers?

Every whole number greater than 1 can be classified as either a prime number or a composite number based on the number of factors it has.

A prime number is a number that has exactly two factors: 1 and the number itself. These are the only numbers that divide the number completely without leaving a remainder. For example, 7 has only two factors: 1 and 7, so it is a prime number. Similarly, 2, 3, 5, 11, and 13 are prime numbers.

A composite number is a number that has more than two factors. It can be divided exactly by numbers other than 1 and itself. For example, 12 has factors 1, 2, 3, 4, 6, and 12, so it is a composite number. Other examples of composite numbers include 4, 6, 8, 9, and 15.
The number 1 is neither prime nor composite because it has only one factor, which is 1. A prime number must have exactly two factors, while a composite number must have more than two factors.

Properties of Prime Numbers

A prime number is a natural number greater than 1 that has exactly two factors: 1 and the number itself.
The smallest prime number is 2. It is also the only even prime number because every other even number has at least three factors: 1, 2, and the number itself.
Examples:
5 has factors 1 and 5, so it is prime.
17 has factors 1 and 17, so it is prime.

Properties of Composite Numbers

A composite number is a natural number greater than 1 that has more than two factors.
Every composite number can be expressed as a product of two or more smaller factors. For example 18 = 2 × 9
Since 18 has factors 1, 2, 3, 6, 9, and 18, it is a composite number.
The smallest composite number is 4 because 4 = 2 × 2 and its factors are 1, 2 and 4
All composite numbers have at least one factor other than 1 and themselves.

Relationship Between Prime and Composite Numbers

Every whole number greater than 1 is either prime or composite.
A prime number has exactly two factors, while a composite number has more than two factors.
For example:
13 is prime because its only factors are 1 and 13.
14 is composite because its factors are 1, 2, 7, and 14.
Prime numbers act as the basic building blocks of composite numbers because every composite number can be written as a product of prime numbers.
For example 60 = 2 × 2 × 3 × 5
Here, 2, 3, and 5 are prime factors of the composite number 60.

Twin Prime Numbers and Prime Pairs

Two prime numbers that differ by exactly 2 are called twin prime numbers.
In other words, if two prime numbers are p and p + 2, they form a twin prime pair.
Examples:
(3, 5) → Difference = 2
(5, 7) → Difference = 2
(11, 13) → Difference = 2
(17, 19) → Difference = 2
Both numbers in a twin prime pair must be prime. A pair of numbers with a difference of 2 cannot be called twin primes if either number is composite.

Co-prime Numbers

Two numbers are called co-prime numbers if their only common factor is 1.
Co-prime numbers do not have to be prime numbers themselves. The condition only depends on their common factors.
Examples:
8 and 15
Factors of 8:
1, 2, 4, 8
Factors of 15:
1, 3, 5, 15
The only common factor is 1, so 8 and 15 are co-prime.

Prime Number Patterns and Properties

Prime numbers follow certain patterns that help in identifying and analysing them.
1. The only even prime number
2 is the only even prime number.
Every other even number is divisible by 2, so it has at least three factors and becomes composite.

2. Last digit property
Except for 2 and 5, every prime number greater than 5 must end in 1, 3, 7, or 9
For example 11, 13, 17, 19, 31, and 47 are possible prime numbers.
However, ending in one of these digits does not guarantee that a number is prime.
For example 21 ends in 1, but it is composite because 21 = 3 × 7

3. Prime numbers and factors
A number is prime if it cannot be divided exactly by any number other than 1 and itself.
For larger numbers, checking divisibility by prime numbers up to the square root of the number is sufficient to determine whether the number is prime.

Summary of Prime and Composite Numbers

Concept	Key Formula / Rule
Prime Number	A number greater than 1 that has exactly two factors: 1 and the number itself.
Composite Number	A number greater than 1 that has more than two factors.
Number 1	1 is neither prime nor composite because it has only one factor.
Smallest Prime Number	The smallest prime number is 2.
Smallest Composite Number	The smallest composite number is 4 because its factors are 1, 2, and 4.
Only Even Prime Number	2 is the only even prime number. Every other even number is composite.
Odd Prime Numbers	All prime numbers greater than 2 are odd numbers.
Factors of a Prime Number	A prime number has only two factors: 1 and itself.
Factors of a Composite Number	A composite number has factors other than 1 and itself.
Prime Factor Relationship	Every composite number can be expressed as a product of prime numbers.
Twin Prime Numbers	Two prime numbers whose difference is exactly 2 are called twin primes.
Examples of Twin Primes	(3, 5), (5, 7), (11, 13), and (17, 19) are examples of twin prime pairs.
Co-prime Numbers	Two numbers are co-prime if their only common factor is 1.
Co-prime Numbers Are Not Necessarily Prime	Two composite numbers can also be co-prime. Example: 8 and 15.
Prime Number Last Digit Rule	Except 2 and 5, prime numbers greater than 5 can only end in 1, 3, 7, or 9.
Checking Prime Numbers	To check whether a number is prime, test divisibility by prime numbers up to its square root.
Classification of Numbers	Every whole number greater than 1 is either prime or composite.

Common Mistakes

Confusing prime numbers with odd numbers. Not every odd number is prime. For example, 9 and 15 are odd numbers but they are composite because they have more than two factors.
Considering 1 as a prime number. The number 1 is neither prime nor composite because it has only one factor.
Forgetting that 2 is the only even prime number. Every other even number is composite because it has at least three factors.
Assuming that a number ending in 1, 3, 7, or 9 is always prime. These digits are possible last digits of prime numbers greater than 5, but numbers such as 21, 33, and 39 are still composite.
Confusing co-prime numbers with prime numbers. Two co-prime numbers only need to have 1 as their common factor. They do not need to be prime. For example, 8 and 15 are co-prime numbers.
Assuming that all pairs of prime numbers are twin primes. Two prime numbers are twin primes only when their difference is exactly 2, such as (11, 13).
Checking only whether the difference between two numbers is 2 while identifying twin primes. Both numbers must also be prime. For example, (9, 11) is not a twin prime pair because 9 is composite.
Forgetting that every composite number has at least one factor other than 1 and itself. A number like 12 is composite because it has factors 2, 3, 4, and 6 in addition to 1 and 12.
Assuming that all numbers with many factors are difficult to classify. A number is composite as soon as it has more than two factors, regardless of how many factors it has.
Checking divisibility by every number when testing whether a number is prime. It is sufficient to check divisibility by prime numbers up to the square root of the given number.

Practice Questions

Question 1: Find the smallest pair of consecutive composite numbers that are also co-prime.

Question 2: A number N is composite and has exactly 4 factors. If two of its factors are 1 and 21, find the possible value of N.

Question 3: Find the smallest composite number that has exactly three factors.

Question 4: A prime number p is greater than 20. When p is increased by 6, the result is also a prime number. Find the smallest possible value of p.

Question 5: Find the number of positive integers n < 30 for which n + 1 is prime and n + 3 is composite.

Question 6: A two-digit number is composite. When its digits are reversed, the new number is prime. Find the smallest possible value of the original number.

Question 7: The sum of two prime numbers is 60. If their difference is also a prime number, find all possible pairs of primes.