Question

Which of the following numbers are squares of even numbers?
$121$ , $225$ , $256$ , $324$ , $1296$ , $6561$ , $5476$ , $4489$ , $373758$

Answer 1

Hint: Here we are given some numbers and we need to obtain the numbers that are squares of even numbers. When we square the even number, we will obtain the even number. For example, we shall square even numbers $2$ . Thus, ${2^2} = 4$
Hence, $4$ is an even number which can be said as $4$ is a square of an even number.

Complete step-by-step answer:
The given numbers are $121$ , $225$ , $256$ , $324$ , $1296$ , $6561$ , $5476$ , $4489$ , $373758$
We need to find the numbers that are squares of even numbers.
a) The given number is $121$
Here the last digit is $1$ which means that the given number $121$ is odd.
When we square an even number, the resultant will be an even number.
But the result of $121$ is odd.
Hence, $121$is not a square of even numbers.
b) The given number is $225$
Here the last digit is five which means that the given number $225$ is odd.
When we square an even number, the resultant will be an even number.
But the result of $225$ is odd.
Hence, $225$is not a square of even numbers.
c) The given number is $256$
Now, we shall use the prime factorization method to find the factors.
$
  2\left| \!{\underline {\,
  {256} \,}} \right. \\
  2\left| \!{\underline {\,
  {128} \,}} \right. \\
  2\left| \!{\underline {\,
  {64} \,}} \right. \\
  2\left| \!{\underline {\,
  {32} \,}} \right. \\
  2\left| \!{\underline {\,
  {16} \,}} \right. \\
  2\left| \!{\underline {\,
  4 \,}} \right. \\
  2\left| \!{\underline {\,
  2 \,}} \right. \\
 $
That is, $256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
$256 = \left( {2 \times 2} \right) \times \left( {2 \times 2} \right) \times \left( {2 \times 2} \right) \times \left( {2 \times 2} \right)$
Here every factor is paired.
Hence $256$is a perfect square.
$256 = {2^2} \times {2^2} \times {2^2} \times {2^2}$
$ \Rightarrow 256 = {16^2}$
Therefore, the given number $256$is a square of an even number.
d) The given number is $324$
Now, we shall use the prime factorization method to find the factors.
$
  2\left| \!{\underline {\,
  {324} \,}} \right. \\
  2\left| \!{\underline {\,
  {162} \,}} \right. \\
  3\left| \!{\underline {\,
  {81} \,}} \right. \\
  3\left| \!{\underline {\,
  {27} \,}} \right. \\
  3\left| \!{\underline {\,
  9 \,}} \right. \\
  3\left| \!{\underline {\,
  3 \,}} \right. \\
 $
That is, $324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3$
$324 = \left( {2 \times 2} \right) \times \left( {3 \times 3} \right) \times \left( {3 \times 3} \right)$
Here every factor is paired.
Hence $324$is a perfect square.
$324 = {2^2} \times {3^2} \times {3^2}$
$ \Rightarrow 324 = {24^2}$
Therefore, the given number $324$is a square of an even number.
e) The given number is $1296$
Now, we shall use the prime factorization method to find the factors.
\[
  2\left| \!{\underline {\,
  {1296} \,}} \right. \\
  2\left| \!{\underline {\,
  {648} \,}} \right. \\
  2\left| \!{\underline {\,
  {324} \,}} \right. \\
  2\left| \!{\underline {\,
  {162} \,}} \right. \\
  3\left| \!{\underline {\,
  {81} \,}} \right. \\
  3\left| \!{\underline {\,
  {27} \,}} \right. \\
  3\left| \!{\underline {\,
  9 \,}} \right. \\
  3\left| \!{\underline {\,
  3 \,}} \right. \\
 \]
That is, $1296 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3$
$1296 = \left( {2 \times 2} \right) \times \left( {2 \times 2} \right) \times \left( {3 \times 3} \right) \times \left( {3 \times 3} \right)$
Here every factor is paired.
Hence $1296$is a perfect square.
$1296 = {2^2} \times {2^2} \times {3^2} \times {3^2}$
$ \Rightarrow 1296 = {36^2}$
Therefore, the given number $1296$is a square of an even number.
f) The given number is $6561$
Here the last digit is $1$ which means that the given number $6561$is odd.
When we square an even number, the resultant will be an even number.
But the result of $6561$ is odd.
Hence, $6561$is not a square of even numbers.
g) The given number is $5476$
Now, we shall use the prime factorization method to find the factors.
$
  2\left| \!{\underline {\,
  {5476} \,}} \right. \\
  2\left| \!{\underline {\,
  {2738} \,}} \right. \\
  37\left| \!{\underline {\,
  {1369} \,}} \right. \\
  37\left| \!{\underline {\,
  {37} \,}} \right. \\
 $
That is, $5476 = 2 \times 2 \times 37 \times 37$
$5476 = \left( {2 \times 2} \right) \times \left( {37 \times 37} \right)$
Here every factor is paired.
Hence $5476$is a perfect square.
$5476 = {2^2} \times {37^2}$
$ \Rightarrow 5476 = {74^2}$
Therefore, the given number $5476$is a square of an even number.
h) The given number is $4489$
Here the last digit is nine which means that the given number $4489$is odd.
When we square an even number, the resultant will be an even number.
But the result of $4489$ is odd.
Hence, $4489$is not a square of even numbers.
i) The given number is $373758$
Now, we shall use the prime factorization method to find the factors.
$
  2\left| \!{\underline {\,
  {373758} \,}} \right. \\
  3\left| \!{\underline {\,
  {186879} \,}} \right. \\
  7\left| \!{\underline {\,
  {62293} \,}} \right. \\
  11\left| \!{\underline {\,
  {8899} \,}} \right. \\
  809\left| \!{\underline {\,
  {809} \,}} \right. \\
 $
That is, $373758 = 2 \times 3 \times 7 \times 11 \times 809$
Here every factor is not paired, appearing only one time.
Hence grouping of factors into pairs is not possible.
Hence, $373758$is not a square of even numbers.
Therefore, $256$ , $324$ , $1296$ and $5476$are the numbers which are the squares of even numbers.

Note: When we square an odd number, the resultant number that we got is also odd. For example, we shall consider an odd number $3$. Now we shall square $3$
Thus, we have ${3^2} = 9$ and we got $9$ which is an odd number.
Thus, $9$is a square of an odd number.