180 x 392

= 2^5 x 3² x 7² x 5¹


To make the number a perfect square, we have make all the powers even.

So since it's divided by n we need to divide. But smallest integer of n means we need to divide by the smallest possible integer number.


Since 3² and 7² are already even powers, we just have to make the power of the prime factors 2 and 5 even.

Since 5 = 5¹ and 1 is odd, we divide by 5 to get rid of the 5.

5¹ ÷ 5 = 5¹-¹ = 5^0 = 1

And divide by 2 so that 2^5 becomes 2⁴
2^5 ÷ 2 = 2^(5 - 1) = 2⁴


(180 x 392) ÷ 5 ÷ 2
= 2^5 x 3² x 7² x 5¹ ÷ 5 ÷ 2
= 2⁴ x 3² x 7²
= (2² x 3 x 7)²
= (4 x 3 x 7)²

Now ÷ 5 ÷ 2 is the same as ÷ 10

So smallest integer n = 10