# NCERT Solutions for Class 6th Mathematics

## Chapter 3. Playing with Numbers

### Exercise 3.5

Exercise 3.5

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

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

**Answer : **False

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

**Answer : **True

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

**Answer : **False

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

**Answer :** True

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

**Answer : **False

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

**Answer : **False

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

**Answer : T**rue

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

**Answer : **True

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

**Answer : **False

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

**Solution:**

There are two different way as follow:

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

**Solution:** 1 and the composite number itself not included in the prime factorization of a composite number.

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

**Solution: **The greatest 4-digit number -

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

**Solution: **

The smallest five diigit number is 10000.

It's tree factor is :

Hence the prime factorisation =

2 × 2 × 2 × 2 × 5 × 5 × 5 × 5

**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. **

**Solution: **

Prime factors of 1729 are 7 × 13 × 19.

**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. Thus, the product must be multiple of 6.

Example:

(i) 2 **×** 3 **×** 4 = 24

(ii) 4 **×** 5 **×** 6 = 120

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

**Solution: **3 + 5 = 8 and 8 is divisible by 4.

5 + 7 = 12 and 12 is divisible by 4.

7 + 9 = 16 and 16 is divisible by 4.

9 + 11 = 20 and 20 is divisible by 4.

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

**Solution:** In expressions (b) and (c), prime factorization has been done.

**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]. **

**Solution: **The prime factorization of 45 = 5 **×** 9 25110 is divisible by 5 as ‘0’ is at its unit place.

25110 is divisible by 9 as sum of digits is divisible by 9.

Therefore, the number must be divisible by 5 **×** 9 = 45

**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. **

**Solution: **No. Number 12 is divisible by both 6 and 4 but 12 is not divisible by 24.

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

**Solution: **The smallest four prime numbers are 2, 3, 5 and 7.

Hence, the required number is 2 **×** 3 **×** 5 **×** 7 = 210