Why is the LCM of two square numbers always another square number?

Answer 1

Lets investigate.

To find an LCM you do a prime factorization and choose the factors, which are not in common raised to an appropriate power to show how many times it occurs in one of the 2 numbers AND use all factors that they do not have in common also to an approp. power.

Ex 1 LCM of 49 and 25

prime factorization yields 5x5 and 7x7

SO LCM will be (5x5)x(7x7) because they have no common factors

which equals 352

Ex 2 LCM of 100 and 36

prime factorization yields 5x5x2x2 and 3x3x2x2

They have common factors of 2x2 so use these but dont repeat them

So LCM = (5x5)x(3x3)x(2x2) = (5x3x2)x(5x3x2) = 302

So every square (A) when factored becomes either (a x a) or (b x b x c x c) etc.

Lets find the LCM of A and N where A and N are squares, but not equal.

Prime factorization A = ( a x a x b x b) and N = ( n x n x p x p x a x a)

LCM will equal (b x b) x ( n x n x p x p) x (a x a) <-- common factors

= (a x b x n x p) x (a x b x n x p)

= (a x b x n x p)2 which is a square.

The rules for forming a LCM force you to make a square because each original number has a prime factorization that has 2 factors, and you must use both or you will not form a LCM.