Using the prime factorisation method, find which of the following numbers are perfect squares:
 
 (i) 441

(ii) 576

(iii) 11025

(iv) 1176

Given:   (i) 441  (ii) 576  (iii) 11025  (iv) 1176

To find: Here we have to find which numbers are perfect squares.

Solution:

A perfect square can always be expressed as a product of pairs of equal factors.

(i)

Prime factorization of 441:

441 = 3 $\times$ 3 $\times$ 7 $\times$ 7

441 = 7 $\times$ 3 $\times$ 7 $\times$ 3

441 = 21 $\times$ 21

441 = (21)²  

So, 441 is a perfect square.

(ii)

Prime factorization of 576:

576 = 8 $\times$ 8 $\times$ 3 $\times$ 3

576 = 8 $\times$ 3 $\times$ 8 $\times$ 3

576 = 24$ \times $ 24 

576 = (24)²

So, 576 is a perfect square.

(iii)

Prime factorization of 1025:

1025 =  7 $\times$ 7 $\times$ 3 $\times$ 3 $\times$ 5 $\times$ 5 

1025 = 7 $\times$ 3 $\times$ 5 $\times$ 7 $\times$ 3 $\times$ 5

1025 = 105 $\times$ 105

1025 = 105² 

So, 11025 is a perfect square.

(iv)

Prime factorization of 1176:

1176 = 7 $ \times $ 7 $ \times $ 3 $ \times $ 2 $ \times $ 2 $ \times $ 2

1176 cannot be expressed as a product of two numbers. Thus, 1176 is not a perfect square.

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Updated on: 10-Oct-2022

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