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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.