## Ask Professor Puzzler

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"I can find the LCM of two numbers (like the LCM of 32 and 20 is 160), but how do you find the LCM if there are variables (like 32x^{2}y and 20y^{3})?"

Before addressing your question, I'd like you to think, for a minute, about how you find the LCM of two numbers. When you were in elementary school, you may have been taught to list multiples until you find one that matches:

32: 32, 64, 96, 128, **160**

20: 20, 40, 60, 80, 100, 120, 140, **160**

This is fine, and is probably one of the easiest ways for children to visualize what it means that a number is the least common multiple. But since you're talking about variables, you're ready to approach LCMs in a slightly different way. If you already do LCMs the way I'm going to show below, please bear with me while I review it.

When I want to find the LCM of two numbers, I generally start by doing a prime factorization of the two numbers. You can use a factor tree, or whatever method you've learned. I get the following result:

32 = 2^{5}

20 = 2^{2} · 5

With this factorization, finding the LCM is a cinch. First, the LCM must have both 2 and 5 as factors. Why? Because it's a multiple of numbers that have 2 or 5 (or both) as factors. So I can say:

LCM = 2^{?} · 5^{?}

What are those question marks for? They're there because even though I know the LCM has both 2 and 5 as factors, I haven't figured out what the exponents of those factors are. It's actually pretty simple to figure out, though; we compare the exponents of each factor in the two prime factorizations, and we take the larger exponent. Thus, since we have 2^{5} and 2^{2}, 5 is the bigger exponent, and since we've just got 5^{1} in the second factorization, that's the exponent we'll use for 5.

LCM = 2^{5} · 5^{1} = 160

Now, maybe you think that process is a lot slower than the other process. Maybe. Or maybe it just seemed that way because I was explaining each step. On the other hand, if you have the numbers 1008 and 1012, finding the LCM by the elementary method is going to be brutal (the result is 255024, and you'll have to list out 253 multiples of 1008 and 252 multiples of 1012).

But whether or not it's "easier" isn't actually the point. The point is that this method helps us answer your question about LCMs involving variables.

What are the prime factorization of 32x^{2}y and 20y^{3}?

32x^{2}y = 2^{5} · x^{2} · y

20y^{3} = 2^{2} · 5 · y^{3}

So the LCM has to contain the factors 2, 5, x, and y. And which exponents do we use? The higher exponent for each factor:

LCM = 2^{5} · 5 · x^{2} · y^{3} = 160x^{2}y^{3}.

As a bonus, this method also allows us to find the LCM of some polynomial expressions. For example, find the LCM of x^{2} + 2x + 1 and x^{2} + 3x + 2. As with the previous problems, we begin by factoring:

x^{2} + 2x + 1 = (x + 1)^{2}

x^{2} + 3x + 2 = (x + 1)(x + 2)

What is the LCM? It's (x + 1)^{2}(x + 2)