Question

Using the prime factorization method, find which of the following numbers are perfect squares:

(i) 5625

(ii) 9075

(iii) 4225

(iv) 1089

Answer 1

(i) Given 5625 A perfect square can always be expressed as a product of pairs of equal factors.Now resolve 5625 into prime factors, we get 5625 = 225 X 25 

= 9 X 25 X 25 

= 3 X 3 X 5 X 5 X 5 X 5 

= 3 X 5 X 5 X 3 X 5 X 5

= 75 X 75 

= (75)²

Hence, 75 is the number whose square is 5625

∴ 5625 is a perfect square.

(ii) Given 9075 A perfect square can always be expressed as a product of pairs of equal factors.Now resolve 9075 into prime factors, we get 9075 = 25 X 363

Again resolve 363 into prime factors we get,

9075 = 25 X 363 

= 25 X 121 X 3 

= 5 X 5 X 11 X 11 X 3 

= 5 X 11 X 5 X 11 X 3 

= 55 X 55 X 3

1176 cannot be expressed as a product of two numbers.

Thus 1176 is not a perfect square.

(iii) Given 4225 A perfect square can always be expressed as a product of pairs of equal factors.Now resolve 4225 into prime factors, we get 4225 = 25 X 169 

= 25 X 13 X 13 

= 5 X 5 X 13 X 13 

= 5 X 13 X 5 X 13

= 65 X 65 

= (65)²

Hence, 65 is the number whose square is 4225

∴ 4225 is a perfect square.

(iv)  Given 1089 A perfect square can always be expressed as a product of pairs of equal factors.Now resolve 1089 into prime factors, we get 1089 = 9 X 121 

= 3 X 3 X 11 X 11 

= 3 X 11 X 3 X 11

= 33 X 33 

= (33)²

Hence, 33 is the number whose square is 1089

∴ 1089 is a perfect square.