Class 6 | Exercise 3.2

1. What is the sum of any two (a) Odd numbers? (b) Even numbers?

(a) Sum of two odd numbers:
Example: \(3 + 5 = 8\), \(7 + 9 = 16\), \(11 + 13 = 24\)
→ All results are even.

(b) Sum of two even numbers:
Example: \(4 + 6 = 10\), \(10 + 12 = 22\), \(2 + 8 = 10\)
→ Always even.

(a) Even (b) Even

2. State whether the following statements are True or False:

(a) The sum of three odd numbers is even. → False
Example: \(3 + 5 + 7 = 15\) (odd)

(b) The sum of two odd numbers and one even number is even. → True
Example: \(3 + 5 + 4 = 12\) (even)

(c) The product of three odd numbers is odd. → True
Example: \(3 \times 5 \times 7 = 105\) (odd)

(d) If an even number is divided by 2, the quotient is always odd. → False
Example: \(8 \div 2 = 4\) (even!)

(e) All prime numbers are odd. → False
2 is prime and even!

(f) Prime numbers do not have any factors. → False
They have exactly two factors: 1 and themselves.

(g) Sum of two prime numbers is always even. → False
Example: \(2 + 3 = 5\) (odd)

(h) 2 is the only even prime number. → True

(i) All even numbers are composite numbers. → False
2 is even but prime, not composite.

(j) The product of two even numbers is always even. → True
Example: \(4 \times 6 = 24\) (even)

3. The numbers 13 and 31 are prime numbers. Both have same digits 1 and 3. Find such pairs of prime numbers up to 100.

We look for 2-digit primes that are reverses of each other and both prime.

• 13 and 31 → both prime ✓

• 17 and 71 → both prime ✓

• 37 and 73 → both prime ✓

• 79 and 97 → both prime ✓

Note: 11 and 11 is the same number (not a pair).
19 and 91 → 91 = 7×13 (not prime) ✗

Pairs: (13, 31), (17, 71), (37, 73), (79, 97)

4. Write down separately the prime and composite numbers less than 20.

Numbers from 1 to 19:

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19

Composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18

Remember: 1 is neither prime nor composite.

5. What is the greatest prime number between 1 and 10?

Prime numbers between 1 and 10: 2, 3, 5, 7

Greatest prime number = 7

6. Express the following as the sum of two odd primes.

(a) 44 → \(3 + 41 = 44\) → both odd primes ✓

(b) 36 → \(5 + 31 = 36\) ✓

(c) 24 → \(5 + 19 = 24\) ✓

(d) 18 → \(5 + 13 = 18\) ✓

Many answers are possible! These are just one valid pair each.

7. Give three pairs of prime numbers whose difference is 2. [These are called twin primes.]

• \(3\) and \(5\) → \(5 - 3 = 2\)

• \(5\) and \(7\) → \(7 - 5 = 2\)

• \(11\) and \(13\) → \(13 - 11 = 2\)

Other examples: (17,19), (29,31), etc.

(3, 5), (5, 7), (11, 13)

8. Which of the following numbers are prime? (a) 23 (b) 51 (c) 37 (d) 26

(a) 23 → only divisible by 1 and 23 → Prime

(b) 51 → \(51 = 3 \times 17\) → Not prime

(c) 37 → no divisors other than 1 and 37 → Prime

(d) 26 → even and >2 → Not prime

Prime numbers: 23, 37

9. Write seven consecutive composite numbers less than 100 so that there is no prime number between them.

We need 7 numbers in a row, all composite.

Look between 89 (prime) and 97 (prime):
90, 91, 92, 93, 94, 95, 96

Check each:
90 = even, 91 = 7×13, 92 = even, 93 = 3×31, 94 = even, 95 = 5×19, 96 = even → all composite!

90, 91, 92, 93, 94, 95, 96

10. Express each as the sum of three odd primes.

(a) 21 → \(3 + 5 + 13 = 21\)

(b) 31 → \(3 + 5 + 23 = 31\)

(c) 53 → \(3 + 7 + 43 = 53\)

(d) 61 → \(3 + 5 + 53 = 61\)

Many combinations exist. All numbers used are odd primes.

11. Write five pairs of prime numbers less than 20 whose sum is divisible by 5.

Sum must end with 0 or 5 → divisible by 5.

• \(2 + 3 = 5\) → divisible by 5

• \(2 + 13 = 15\) → divisible by 5

• \(3 + 7 = 10\) → divisible by 5

• \(7 + 13 = 20\) → divisible by 5

• \(3 + 17 = 20\) → divisible by 5

(2,3), (2,13), (3,7), (7,13), (3,17)

12. Fill in the blanks:

(a) A number which has only two factors is called a prime number.

(b) A number which has more than two factors is called a composite number.

(c) 1 is neither prime nor composite.

(d) The smallest prime number is 2.

(e) The smallest composite number is 4.

(f) The smallest even number is 2.