Least Common Multiple (LCM)
The concept of **least common multiple** (**LCM**) can be very useful for many types of questions on the GRE, including finding a common denominator when adding or subtracting fractions with different denominators.
Based upon the name, what do you think the LCM of two integers means?
Not so.
The smallest positive number that divides evenly into two integers will always be 1. The LCM of two integers is the smallest positive number that is a multiple of both of those integers.
As an example, 12 is the LCM of 4 and 6 because it is the smallest positive number that is a multiple of both 4 and 6.
Not so.
The smallest positive number that can be a remainder will always be 1. The LCM of two integers is the smallest positive number that is a multiple of both of those integers.
As an example, 12 is the LCM of 4 and 6 because it is the smallest positive number that is a multiple of both 4 and 6.
That's exactly right!
As an example, 12 is the LCM of 4 and 6 because it is the smallest positive number that is a multiple of both 4 and 6.
The simplest way to find the LCM of two integers is to begin listing the positive multiples of each integer. The smallest number that appears on both lists is the LCM of those integers.
Using this strategy, what is the LCM of 10 and 12?
Not quite.
While 0 is a multiple of all integers, the LCM must always be a positive number, and 0 is not positive.
That's it!
The list of positive multiples of 10 starts 10, 20, 30, 40, 50, __60__, 70, 80, ....
The list of positive multiples of 12 starts 12, 24, 36, 48, __60__, 72, 84, 96, ....
The smallest number on both lists is 60, so this is the LCM of 10 and 12.
Not quite.
While 120 is a multiple of both 10 and 12, the LCM is the smallest positive number that is a multiple of both.
Sometimes, the LCM of two numbers is their product, but in this case there is a smaller positive number that is a multiple of both.
This strategy extends to finding the LCM of more than two integers.
What is the LCM of 3, 6, and 12?
Incorrect.
Three is actually a factor of all of these integers, not a multiple. The LCM of integers cannot be smaller than some of those integers.
That's right!
Twelve is a multiple of itself, as well as 3 and 6. Notice that 12 is both the LCM and one of the original integers on the list.
It is true that if a positive integer is a multiple of the other integers for which you are finding the LCM, then that positive integer is the LCM.
Not quite.
While 36 is a common multiple of 3, 6 and 12, it is not the smallest.
To sum up:
[[summary]]
The smallest positive number that divides evenly into both of those integers
The smallest positive number that can be a remainder when dividing into both of those integers
The smallest positive number that is a multiple of both of those integers
0
60
120
3
12
36
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