In yesterday’s post on the LCM, I wrote about 375 pages on the topic, and then I said that I left out an idea. Hahaha, you probably thought. Very funny, Josh.
But never fear. I am not going to write another 375 pages on the topic.
What I do need to bring to your attention, though, is that there are two LCM situations that I did not take into account yesterday. So to present a complete picture, I need to explain (for those who have not already figured this out by themselves) how to use my new technique in those two situations.
You will notice that in my write-up yesterday — and in the practice problems I provided — the gap always divided evenly into the smaller number. How convenient, right? In the first example, we had a gap of 3 dividing into 12; in the next, a gap of 4 going into 20. Of course this does not always happen. Consider a situation in which we want to find the LCM for 10 and 16. The gap of 6 (16 – 10 = 6) does NOT divide evenly into the smaller number, 10. So what would we do here?
In such a situation, we’d run through the multiples of the smaller number till we hit a multiple that the gap does divide into evenly. In this example of 10 and 16, we would look at the first multiples of 10: 20, 30, 40, 50, etc., till we find one that 6 does divide into evenly. Whoa! No need to go even that far, you point out … since 6 does divide evenly into 30. How many times? Five times. After getting this number, 5, follow the same procedure I laid out yesterday … multiply this quotient, 5, by the larger number, 16, to get the LCM. So the LCM would be 80, since 5 x 16 = 80.
For practice, look at one more situation: find the LCM for 25 and 35. The gap, 10 (35 – 25 = 10), does NOT divide evenly into the smaller number, 25. So we would check out the multiples of 25: 25, 50 … BINGO! 10 does divide evenly into 50. How many times? 5. So we would just multiply 5 by the larger number, 35, getting 175. And that is the LCM for 25 and 35.
Note that in both of these number pairs: 10 and 16, and 25 and 35, there is a common factor for each pair. For 10 and 16 the common factor is 2; for 25 and 35, the common factor is 5. The fact that such a common factor exists for each pair is what allows us to use this approach. The common factor is also what makes it work out that the LCM is less than the mere product of the two numbers.
But as you may recall from studying this topic in other places, whenever two numbers do NOT have a common factor, the numbers are said to be “relatively prime.” And whenever two numbers are relatively prime, their LCM is nothing less than the product of the numbers.
Example: 9 and 16. Each number is composite, yet they share no common factors. The non-trivial factors of 9 are 3 and 9; the non-trivial factors of 16 are 2, 4, 8, and 16. No factors overlap, so 9 and 16 are relatively prime. That being the case the LCM for 9 and 16 is simply the product: 9 x 16 = 144.
And this is indeed the case whenever two numbers are relatively prime.
For our purposes, what we need to keep in mind is this: we can use the new technique I have presented yesterday and today whenever two numbers have a common factor. But when the two numbers do NOT have a common factor, we cannot use this technique. Instead, we just multiply the two numbers together, and the product we get is the LCM.
And now, some practice problems to help you nail down this method for finding the LCM.
For the following problems, first determine whether or not the numbers are relatively prime. If they’re relatively prime, multiply them to find the LCM. If they’re not relatively prime, use the method outlined in this post to find the LCM. The answers follow.
a) 6 and 10
b) 8 and 15
c) 8 and 14
d) 12 and 20
e) 27 and 35
f) 15 and 25
g) 32 and 55
h) 28 and 36
i) 64 and 77
j) 48 and 58
a) 6 and 10 — LCM = 30
b) 8 and 15 — Relatively prime, LCM = 120
c) 8 and 14 — LCM = 56
d) 12 and 20 — LCM = 60
e) 27 and 35 — Relatively prime, LCM = 945
f) 15 and 25 — LCM = 75
g) 32 and 55 — Relatively prime, LCM = 1,760
h) 28 and 36 — LCM = 252
i) 64 and 77 — Relatively prime, LCM = 4,928
j) 48 and 58 — LCM = 1,392
Josh Rappaport is the author of the award-winning Algebra Survival Guide and several other supplemental math books. To check out these products, follow the links in the sidebar, or just visit the Singing Turtle Press website here.