How do you find the LCM of 35, 25?

You could think of it like this: the multiples of each of these factors are:
#25: 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325, 350...#
#35: 35, 70, 105, 140, 175, 210, 245, 280, 415, 350...#

But that takes too long. Also, notice that while 350 IS a common multiple of both 25 and 35, it is not the LEAST common multiple. There is a smaller number that both of our factors, 25 and 35, go into evenly. If you look, you can see in the lists that 175 is that number.

So how do you get that without making long lists?

First, you have to find the "prime factorization" of each of your factors. That means breaking each factor down into prime numbers and prime numbers only (remember, 1 is neither prime nor composite).

So: #25 = 5 xx 5.# Thus,# 5 xx 5, or 5^2# is the prime factorization of 25.

And: #35 = 5 xx 7.# Thus, #5 xx 7# is the prime factorization of 35.

Next, you have to remember that BOTH of your factors have to go into the LCM exactly, or it won't work!

So, how many 5s do we need to allow 25 to go into whatever the LCM is? Two.

How many 5s do we need to allow 35 to go into whatever the LCM is? One

Since we are looking for the LEAST common multiple, we will use the two 5s from 25. There is no need to use the 5 from the 35, since that is already covered.

So, how many 7s do we need to allow 25 to go into the LCM? none. How many 7s do we need to allow 35 to go into the LCM? one.
Therefore, we must use one 7.

This brings us to the final step:
We need to use the two 5s and the one 7 and multiply them together.

#5 xx 5 xx 7 = 175#, and that is our LCM.

You can easily check this by noticing that you can "find" 25 in the final expression - it's #5xx5#! And, you can easily "find" 35 in the final expression - it's #5xx7#!

So, you know that both 25 and 35 are factors of - and will go exactly into - the LCM of 175.