What is the smallest counting number which when multiplied to 2340 will make it a perfect square

First, factor 2340

2340 =>
234 * 10 =>
2 * 117 * 2 * 5 =>
2 * 9 * 13 * 2 * 5 =>
2^2 * 3^2 * 5 * 13

A perfect square is made up of factors like this:

a^2 * b^2 * c^2 * d^2 * ....

So, in order to get that, we need another 5 and 13

5 * 13 = 65

65 is the answer.

The prime factorization of 2340 is:

2340 = 2 2 3 3 5 13

If we want the product of this and some other number to be a perfect square, then the product must contain an even number of each prime factor. The smallest such number is therefore:

5 x 13 = 65

And 65 x 2340 = 390 = 2 3 5 13

Trying division by 1, 4, 9, 16, 25, 36, etc as follows :
2340 / 4 = 585
585 / 9 = 65
Multiply 2340 by 65 to make it a perfect Square number. ANSWER

factor 2340=4*585=4*5*117=4*3*5*39=4*9*5*13=2^2…

So multiply by 5*13 (so every prime is to an even power!)

prime 2340
u l need 65 to make it perfect square