Question

Using prime factorization method, find which of the following numbers are perfect square?
$189,225,2048,343,441,2916,11025,3549$

Answer 1

The given numbers are $189,225,2048,343,441,2916,11025,3549$.

$\left(1\right)$ Representing $189$ as the product of its prime factors,

$189=3^2\times 3\times 7$

Since, it does not have equal pairs of factors, it is not a perfect square.

$\therefore$ $189$ is not a perfect square.

$\left(2\right)$ Representing $225$ as the product of its prime factors,

$225=(5\times 5)\times (3\times 3)$

Since $225$ has equal pairs of factors, it is a perfect square.

$\therefore 225$ is a perfect square.

$\left(3\right)$ Representing $2048$ as the product of its prime factors,

$2048=(2\times 2)\times (2\times 2)\times (2\times 2)\times (2\times 2)\times (2\times 2)\times 2$

Since $2048$ does not have equal pairs of factors, it is not a perfect square.

$\therefore 2048$ It is a not a perfect square.

$\left(4\right)$ Representing $343$ as the product of its prime factors,

$343=(7\times 7)\times 7$

Since $343$ does not have equal pairs of factors, it is not a perfect square.

$\therefore 343$ is not a perfect square.

$\left(5\right)$ Representing $441$ as the product of its prime factors,

$441=(7\times 7)\times (3\times 3)$

Since $441$ has equal pairs of factors, it is a perfect square.

$\therefore 441$ is a perfect square.

$\left(6\right)$ Representing $2916$ as the product of its prime factors,

$2916=(3\times 3)\times (3\times 3)\times (3\times 3)\times (2\times 2)$

Since $2961$ has equal pairs of factors, it is a perfect square.

$\therefore 2916$ is a perfect square.

$\left(7\right)$ Representing $11025$ as the product of its prime factors,

$11025=(3\times 3)\times (5\times 5)\times (7\times 7)$

Since $11025$ has equal pairs of factors, it is a perfect square.

$\therefore 11025$ is a perfect square.

$\left(8\right)$ Representing $3549$ as the product of its prime factors,

$3549=(13\times 13)\times 3\times 7$

Since $3549$ does not have equal pairs of factors, it is not a perfect square.

$\therefore 3549$ It is a not a perfect square.

Hence, out of all the given numbers, $225,441,2916and11025$ are perfect squares.