NCERT Solutions Class 6th Maths Chapter – 3 Playing With Numbers Exercise – 3.5

Question 1. Which of the following statements are true?

(a) If a number is divisible by 3, it must be divisible by 9.
Solution – False

(b) If a number is divisible by 9, it must be divisible by 3.
Solution – True

(c) A number is divisible by 18, if it is divisible by both 3 and 6.
Solution – False

(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.
Solution – True

(e) If two numbers are co-primes, at least one of them must be prime.
Solution – False

(f) All numbers which are divisible by 4 must also be divisible by 8.
Solution – False

(g) All numbers which are divisible by 8 must also be divisible by 4.
Solution – True

(h) If a number exactly divides two numbers separately,- it must exactly divide their sum.
Solution – True

(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.
Solution – False

Question 2. Here are two different factor trees for 60. Write the missing numbers.

(a)

Solution –

Given that-

Here, 6 = 2 x missing number
∴ Missing number = 6 ÷ 2 = 3
Similarly, 10 = 5 x missing number
∴ Missing number = 10 ÷ 5 = 2
Hence, the missing numbers are 3 and 2.

(b)

Solution –

Given that-

Let the missing numbers be m₁, m₂, m₃ and m₄.
60 = 30 x m₁
⇒ m₁ = 60 ÷ 30 = 2
30 = 10 x m₂
⇒ m₂ = 30 ÷ 10 = 3
10 = m₃ x m₄
⇒ m₃ = 2 or 5 and m₄ = 5 or 2
Hence, the missing numbers are 2, 3, 2, 5.

Question 3. Which factors are not included in the prime factorisation of a composite number?

Solution –

1 and the number itself are not included in the prime factorisation of a composite number.

Question 4. Write the greatest 4-digit number and express it in terms of its prime factors.

Solution –

The greatest 4-digit number = 9999

Hence, the prime factors of 9999 = 3 x 3 x 11 x 101

Question 5. Write the smallest 5-digit number and express it in the form of its prime factors.

Solution –

The smallest 5-digit number = 10000

Hence, the required prime factors: 10000 = 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5

Question 6. Find all the prime factors of 1729 and arrange them in ascending order. Now state the relations, if any, between the two consecutive prime factors.

Solution –

Given number = 1729

Hence, the prime factors of 1729 = 7 x 13 x 19
Here, 13 – 7 = 6 and 19 – 13 = 6
We see that the difference between two consecutive prime factors is 6

Question 7. The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.

Solution –

Among the three consecutive numbers, there must be one even number and one multiple of 3.
As 6 = 2 x 3, thus the product must be multiple of 6.
Example:
(i) 2 × 3 × 4 = 24
(ii) 4 × 5 × 6 = 120

Question 8. The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.

Solution –

The sum of any two consecutive numbers is always odd, and the sum of two consecutive odd numbers is always divisible by 4.

= 13 + 15 = 28 and 28 is divisible by 4
= 5 + 7 = 12 and 12 is divisible by 4
= 17 + 19 = 36 and 36 is divisible by 4
= 9 + 11 = 20 and 20 is divisible by 4

Question 9. In which of the following expressions, prime factorization has been done?

(a) 24 = 2 x 3 x 4

Solution –

24 = 2 x 3 x 4
Here, 4 is not a prime number.
Hence, 24 = 2 x 3 x 4 is not a prime factorization.

(b) 56 = 7 x 2 x 2 x 2

Solution –

56 = 7 x 2 x 2 x 2
Here, all factors are prime numbers
Hence, 56 = 7 x 2 x 2 x 2 is a prime factorization.

(c) 70 = 2 x 5 x 7

Solution –

70 = 2 x 5 x 7
Here, all factors are prime numbers.
Hence, 70 = 2 x 5 x 7 is a prime factorization.

(d) 54 = 2 x 3 x 9

Solution –

54 = 2 x 3 x 9
Here, 9 is not a prime number.
Hence, 54 = 2 x 3 x 9 is not a prime factorization.

Question 10. Determine if 25110 is divisible by 45.

Solution –

45 = 5 x 9
Here, 5 and 9 are co-prime numbers.
Test of divisibility by 5- unit place of the given number 25110 is 0. So, it is divisible by 5.
Test of divisibility by 9-
Sum of the digits = 2 + 5 + l + l + 0 = 9 which is divisible by 9.
So, yes the given number is divisible by 5 and 9 both. Hence, the number 25110 is divisible by 45.

Question 11. 18 is divisible by both 2 and 3. It is also divisible by 2 x 3 = 6. Similarly, a number is divisible by both 4 and 6. Can we say that the number must also be divisible by 4 x 6 = 24? If not, give an example to justify your answer.

Solution –

No, Number 12 is divisible by both 4 and 6; but 12 is not divisible by 24.

Question 12. I am the smallest number, having four different prime factors. Can you find me?

Solution –

We know that the smallest 4 prime numbers are 2, 3, 5 and 7
Hence, the required number = 2 x 3 x 5 x 7 = 210.