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Using prime factorization method, find which of the following numbers are perfect squares?
(i) 189
(ii) 225
(iii) 2048
(iv) 343
(v) 441
(vi) 2916
(vii) 11025
(viii) 3549
To do :
We have to find whether the given numbers are perfect squares using the prime factorization method.
Solution:
Perfect Square: A perfect square has each distinct prime factor occurring an even number of times.
(i) $189=3\times3\times3\times7$
$=3\times(3)^2\times7$
189 does not have distinct prime factors occurring even number of times.
Therefore, 189 is not a perfect square.
(ii) $225=3\times3\times5\times5$
$=(3)^2\times(5)^2$
225 has distinct prime factors occurring an even number of times.
Therefore, 225 is a perfect square.
(iii) $2048=2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2$
$=(2)^2\times(2)^2\times(2)^2\times(2)^2\times(2)^2\times2$
2048 does not have distinct prime factors occurring even number of times.
Therefore, 2084 is not a perfect square.
(iv) $343=7\times7\times7$
$=(7)^2\times7$
343 does not have distinct prime factors occurring even number of times.
Therefore, 343 is not a perfect square.
(v) $441=3\times3\times7\times7$
$=(3)^2\times(7)^2$
441 has distinct prime factors occurring an even number of times.
Therefore, 441 is a perfect square.
(vi) $2916=2\times2\times3\times3\times3\times3\times3\times3$
$=(2)^2\times(3)^2\times(3)^2\times(3)^2$
2916 has distinct prime factors occurring an even number of times.
Therefore, 2916 is a perfect square.
(vii) $11025=3\times3\times5\times5\times7\times7$
$=(3)^2\times(5)^2\times(7)^2$
11025 has distinct prime factors occurring an even number of times.
Therefore, 11025 is a perfect square.
(viii) $3549=3\times7\times13\times13$
$=3\times7\times(13)^2$
3549 does not have distinct prime factors occurring even number of times.
Therefore, 3549 is not a perfect square.