Question

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

(i) 441

(ii) 576

(iii) 11025

(iv) 1176

Answer 1

(i) Given 441, A perfect square can always be expressed as a product of pairs of equal factors.Now resolve 441 into prime factors, we get 441 = 49 X 9 

= 7 X 7 X 3 X 3 

= 7 X 3 X 7 X 3 

= 21 X 21 

= (21)²

Hence, 21 is the number whose square is 441

∴ 441 is a perfect square.

(ii) Given 576 A perfect square can always be expressed as a product of pairs of equal factors.Now resolve 576 into prime factors, we get 576 = 64 X 9 

= 8 X 8 X 3 X 3 

= 8 X 3 X 8 X 3 

= 24 X 24 

= (24)²

Hence, 24 is the number whose square is 576

∴ 576 is a perfect square.

(iii) Given 11025 A perfect square can always be expressed as a product of pairs of equal factors.Now resolve 11025 into prime factors, we get 11025 = 441 X 25 

= 49 X 9 

= 7 X 7 X 3 X 3 X 5 X 5 

= 7 X 3 X 5 X 7 X 3 X 5 

= 105 X 105 

= (105)²

Hence, 105 is the number whose square is 11025

∴ 11025 is a perfect square.

(iv) Given 1176 A perfect square can always be expressed as a product of pairs of equal factors.Now resolve 1176 into prime factors, we get 1176 = 7 X 168

Again resolve 168 into prime factors we get,

1176 = 7 X 168 

= 7 X 21 X 8 

= 7 X 7 X 3 X 2 X 2 X 2

1176 cannot be expressed as a product of two numbers.

Thus 1176 is not a perfect square.