Question

Find the number(s) that can be multiplied with $672$ to make it a perfect square.

• A. $24$
• B. $42$
• C. $168$
• D. $186$

Answer 1

Answer: (B), (C)

$168$

The sum of the digits of $672$ is divisible by 3.
$\therefore$ 672 is divisible by 3.
Also, the last two digits are divisible by 4.
Hence, it is divisible by $4.$

As it is divisible by 3 and 4 so, 672 must be divisible by 12.

$\Rightarrow \frac{672}{12}=56=7\times 8$

$\therefore 672=12\times 7\times 8$
$\Rightarrow 672=4\times 3\times 7\times 2^3$
$\Rightarrow 672=2^2\times 2^3\times 3\times 7$
$\Rightarrow 672=(2^2{)}^2\times 2\times 3\times 7$

Here, $2\times 3\times 7$ is not making a perfect square.
$\therefore$ We need to multiply $2\times 3\times 7=42$ to make it a perfect square.

However, $42\times 2^2,$ i.e., $168$ is also a possible option.