If you take any given number N, and factor it, you may or may not have squares in its root... although actually, you always do. And using these "square factors" are how bringing any number N "into square" (including determining all the multiples that will do so, and in any type of progression) is done:
If you "do not have a square in the root", then the smallest next square is to multiply by itself (because 1 is the only square number in the root and you always have that).
If you do have a square number in its root, then divide and/or multiply your number by these will yield the multiples that will bring N "into square". You can obtain progression by writing a rule that divides by the largest (to get the smallest multiple number), then the 2nd largest, etc..... on through multiplying by the smallest, and up through multiplying by the largest.
For example, 20: 20 roots into 5, 2, 2, 1 and 4 is a square number. So 20/4 = 5 and 20*4 = 80, which when multiplied by 20 yields 100 and 1600. The progression of multiples for 20 that will yield square numbers are 5, 20, 80.
Another example, 36: 36 roots into 3, 3, 2, 2, 1 So 36/9 = 4, 36/4 = 9, 36*4 = 144, and 36*9 = 324. So multiples for 36 that will result in square numbers once multiplied are 4, 9, 36, 144, 324 with resultant square numbers being 144, 324, 1296, 5184, 11664. Important note: using the numbers own square root (in cases where the original N is square, as with 36) does not work (36/6 = 6 and 36*6 does NOT yield a square number and this is because 6 in itself is not a square number).
I have a website where I came across the need to find such multiples for 6808, and it held true with the square in the root being 4. So multiples were 1702, 6808, and 27232. Brute force in Excel did verify that 1702 was indeed the smallest multiple to bring 6808 into square (result = 11587216, square root of which is 3404).
In a nutshell, it is an extension of the fact that any square number multiplied by any square number will yield a square number. But by using the "root factors that are square" to derive your multiples instead, you expand the ability into being able to convert ANY number into a square, through all of its possible progressions --- without brute force.