Question

Find the smallest number by which the given number must be multiplied so that the product is a perfect square: 

(i) 23805 

(ii) 12150 

(iii) 7688

Answer 1

(i) 23805 

Resolving 23805 into prime factors, we get 

23805 = 3 × 3 × 23 × 23 × 5 

Obtained factors can be paired into equal factors except for 5 

To pair it equally multiply with 5 

23805 × 5 = 3 × 3 × 5 × 5 × 23 × 23 

Again, 

23805 × 5 = (3× 5 × 23) × (3 × 5 × 23) 

= 345 × 345 

= (345)² 

Therefore, product is the square of 345.

(ii) 12150 

Resolving 12150 into prime factors, we get

12150 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 × 2 

Obtained factors can be paired into equal factors except for 2 

To pair it equally multiply with 2 

12150 × 2 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 3 × 3

 Again, 

12150 × 2 = (5 × 3 × 2 × 2 × 2) × (5 × 3 × 2 × 2 × 2)

= 120 × 120 

= (120)² 

Therefore, product is the square of 120.

(iii) 7688 

Resolving 7688 into prime factors, we get 

7688 = 2 × 2 × 31 × 31 × 2 

Obtained factors can be paired into equal factors except for 2 

To pair it equally multiply with 2 

7688 × 2 = 2 × 2 × 2 × 2 × 31 × 31 

Again, 

7688 × 2 = (2× 2 × 31) × (2 × 2 × 31) 

= 124 × 124 

= (124)² 

Therefore, product is the square of 124.