# Which of the following numbers are perfect squares ?(i) 484(ii) 625(iii) 576(iv) 941(v) 961(vi) 2500.

To do :

We have to find whether the given numbers are perfect squares.

Solution:

Perfect Square: A perfect square has each distinct prime factor occurring an even number of times.

(i) Prime factorisation of 484 $=2\times2\times11\times11$

484 has distinct prime factors occurring an even number of times.

Therefore, 484 is a perfect square.

(ii) Prime factorisation of 625 $=5\times5\times5\times5$

625 has distinct prime factors occurring an even number of times.

Therefore, 625 is a perfect square.

(iii) Prime factorisation of 576 $=2\times2\times2\times2\times2\times2\times3\times3$

576 has distinct prime factors occurring an even number of times.

Therefore, 576 is a perfect square.

(iv) Prime factorisation of 941 $=941\times1$

941 has no prime factors occurring even number of times.

Therefore, 941 is not a perfect square.

(v) Prime factorisation of 961 $=31\times31$

961 has distinct prime factors occurring an even number of times.

Therefore, 961 is a perfect square.

(vi) Prime factorisation of 2500 $=2\times2\times5\times5\times5\times5$

2500 has distinct prime factors occurring an even number of times.

Therefore, 2500 is a perfect square.