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Toggle## Class 6, Maths, Chapter 3, Exercise 3.5 Solutions

**Q.1. Which of the ****following statements are true?**

**(a) If a number is divisible by 3, it must be divisible by 9.**

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

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

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

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

**(f) All numbers which are divisible by 4 must also be divisible by 8.**

**(g) All numbers which are divisible by 8 must also be divisible by 4.**

**(h) If a number exactly divides two numbers separately, it must exactly divide their sum.**

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

**Ans:**

**(a) False,** 6 is divisible by 3 but is not divisible by 9

(b) **True,** as 9 = 3 × 3. Hence, if a number is divisible by 9, it will also be divisible by 3.

For example: Example: 27 is divisible by both 9 and 3

(c) **True,** 18 is the product of 3 and 6, so any number that is divisible by both 3 and 6 must also be divisible by 18.

Example: 36 is divisible by 18 and by both 3 and 6, because 3 and 6 are factors of 18

(d) **True**, 90 is the least common multiple (LCM) of 9 and 10, so any number that is divisible by both 9 and 10 must also be divisible by 90.

(e) **False**, Since 15 and 32 are co-primes and also composite numbers

(f) **False,** For example, 4 is divisible by 4, but not by 8.

(g) **True**, as 2 × 4 = 8. Any number that is divisible by 8 must also be divisible by 4, since 8 is the product of 4 and 2.

(h) **True**, as 2 divides 4 and 8 and it also divides 12 (4 + 8 = 12)

(i) **False**, For example, 5 exactly divides the sum of 3 and 7, but it does not exactly divide either 3 or 7 separately.

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

**Ans:**

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

**Ans: **

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

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

**Ans: **The greatest 4- digit number is 9999

Therefore the prime factors of 9999 = 3 × 3 × 11 × 101

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

**Ans:**

The smallest 5 digit number = 10000

The prime factors of 10000:

= 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5

**Q.6. Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors.**

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

** ****Ans:**

Let’s take some examples to verify the statement “The product of three consecutive numbers is always divisible by 6”.

**Example 1:**

Let’s take the consecutive numbers 2, 3, and 4.

The product of these numbers is: 2 x 3 x 4 = 24.

We can see that 24 is divisible by 6 (24 ÷ 6 = 4).

**Example 2:**

Let’s take the consecutive numbers 5, 6, and 7.

The product of these numbers is: 5 x 6 x 7 = 210.

We can see that 210 is divisible by 6 (210 ÷ 6 = 35).

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

**Ans:**

(i)** **3 + 5 = 8 which is divisible by 4

(ii) 5 + 7 = 12 which is divisible by 4

(iii) 7 + 9 = 16 which is divisible by 4

Therefore, the sum of two consecutive odd numbers is divisible by 4.

**Q.9. In which of the following expressions, prime factorisation has been done?**

**(a) 24 = 2 × 3 × 4**

**(b) 56 = 7 × 2 × 2 × 2**

**(c) 70 = 2 × 5 × 7 **

**(d) 54 = 2 × 3 × 9**

**Ans:**

**The correct answer is (c) 70 = 2 × 5 × 7.**

Since, all the factors are prime. Hence, prime factorisation has been done.

**Explanation:**

**(a) ****24 = 2 × 3 × 4**

Since, 4 is composite. Hence, prime factorisation has not been done.

**(b) 56 = 7 × 2 × 2 × 2**

Since, all the factors are already expressed in prime factorisation form. Hence, prime factorisation has been done.

**(d) 54 = 2 × 3 × 9**

Since, 9 is composite. Hence prime factorisation has not been done.

**In other word,**

It is not a prime factorisation because 9 is not a prime number. It can be simplified to 54 = 2 × 3 × 3 × 3 .

**Q.10. Determine if 25110 is divisible by 45.**

**[Hint : 5 and 9 are co-prime numbers. Test the divisibility of the number by 5 and 9].**

**Ans***: *

*If the last digit of the number 0, then it is divisible by 5.*

*if the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9*

The prime factorization of 45 = 5 x 9

The last digit of 25110 is 0. Hence, it is divisible by 5

25110 is divisible by 9 as sum of digits (2+5+1+1+0 = 9) is divisible by 9.

Since the number is divisible by both 5 and 9

Therefore 25110 is divisible by 45.

**Alternatively:**

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

**Ans:**

No, since, 12 and 36 are both divisible by 4 and 6. But 12 and 36 are not divisible by 24

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

**Ans: **

Yes, here the smallest number with four different prime factors is:

= 2 x 3 x 5 x 7

= 210.

Here, 2, 3, 5, and 7 are all different prime factors, and 210 is the smallest number having four different prime factors.

## NCERT Solutions For Class 6 Maths, Chapter 3 Playing With Numbers (All Exercises)

**Class 6, Maths, Chapter 3, Playing With Numbers **

**Class 6, Maths, Chapter 3, Playing With Numbers, Exercise 3.1**

**Class 6, Maths, Chapter 3, Playing With Numbers, Exercise 3.2**

**Class 6, Maths, Chapter 3, Playing With Numbers, Exercise 3.3**

**Class 6, Maths, Chapter 3, Playing With Numbers, Exercise 3.4**

**Class 6, Maths, Chapter 3, Playing With Numbers, Exercise 3.5** **← You are here**

**Class 6, Maths, Chapter 3, Playing With Numbers, Exercise 3.6**

**Class 6, Maths, Chapter 3, Playing With Numbers, Exercise 3.7**