NCERT Solutions Class 6 Maths Chapter 3 Playing with Numbers Exercise 3.5

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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.