Lesson: Least Common Multiple (LCM)

Great question. Here are two different approaches:

Approach #1: https://gmatclub.com/forum/if-x-is-a-positive-integer-what-is-the-least-...

Approach #2: https://gmatclub.com/forum/if-x-is-a-positive-integer-what-is-the-least-...

Please let me know if either of those approaches help.

Cheers,
Brent

Question link: https://gmatclub.com/forum/the-number-of-students-who-attend-a-school-co...

The technique for finding the LCM of 3 numbers is a little bit different from finding the LCM of 2 numbers.
10 = (2)(5)
12 = (2)(2)(3)
16 = (2)(2)(2)(2)

The three numbers share ONE 2 in their prime factorizations.
So far, LCM = (2)(some other stuff)

Ignore those three shared 2's to get:
10 = (-)(5)
12 = (-)(2)(3)
16 = (-)(2)(2)(2)

Among the remaining primes, two numbers share ONE 2
So, we now have: LCM = (2)(2)(some other stuff)

Ignore those two shared 2's to get:
10 = (-)(5)
12 = (-)(-)(3)
16 = (-)(-)(2)(2)

Among the remaining primes, none are shared.
So we take these remaining primes and add them to our calculations to get:
LCM = (2)(2)(5)(3)(2)(2) = 240

-------ALTERNATE APPROACH----------------
When it comes to finding the LCM of three integers, I prefer to just "build" the LCM by examining one number at a time.
Let N = the LCM of 10, 12 and 16
This means N is divisible by 10, 12 and 16

10 = (2)(5)
So, if N is divisible by 10, there must be one 2 and one 5 in the prime factorization of N.
N = (2)(5)(more possible primes)

12 = (2)(2)(3)
So, if N is divisible by 12, there must be two 2's and one 3 in the prime factorization of N.
We already have ONE 2 in the prime factorization of N. So, we need only add one more 2 (and one 3)
We get: N = (2)(5)(2)(3)(more possible primes)

16 = (2)(2)(2)(2)
So, if N is divisible by 16, there must be four 2's in the prime factorization of N.
We already have TWO 2's in the prime factorization of N, we need to add TWO more 2's
We get: N = (2)(5)(2)(3)(2)(2)

Done!

Here's another way to look at it:
If some integer N is the LCM of 2, 3, 4, 5, 6, 7, 8, and 9, then the prime factorization of N must include 2, 3, 4, 5, 6, 7, 8, and 9

So, for example, if and is divisible by 2, then there must be a 2 hiding in the prime factorization of N.
In other words, we now know that N = (2)(?)(?)(?)(?), where the other "?" symbols represent other possible values in the prime factorization of N.

Next, there must be a 3 hiding in the prime factorization of N.
So we now know that N = (2)(3)(?)(?)(?)(?)

Important: In order for N to be divisible by 4, there must be two 2's hiding in the prime factorization of N.
Since we already have one 2 in the prime factorization of N, we need only add one more 2.
In other words: N = (2)(3)(2)(?)(?)(?)(?)
As we can see, we can't be certain that N is divisible by 2, 3 and 4

Next, there must be a 5 hiding in the prime factorization of N.
So we now know that N = (2)(3)(2)(5)(?)(?)(?)

6 = (2)(3)
So, in order for N to be divisible by 6, there must be one 2 and one 3 hiding in the prime factorization of N.
Since our prime factorization of N already includes one 2 and one 3, we need not add any numbers to our prime factorization.
In other words, if N = (2)(3)(2)(5), then we can be certain that N is divisible by 6.

Next, there must be a 7 hiding in the prime factorization of N.
So we now know that N = (2)(3)(2)(5)(7)(?)(?)

8 = (2)(2)(2)
So, in order for N to be divisible by 8, there must be three 2's hiding in the prime factorization of N.
Since our prime factorization of N already includes two 2's, we need to add one more 2.
In other words, if N = (2)(3)(2)(5)(7)(2)(?)(?), then we can be certain that N is divisible by 8.

9 = (3)(3)
So, in order for N to be divisible by 9, there must be two 3's hiding in the prime factorization of N.
Since our prime factorization of N already includes one 3, we need to add one more 3.
In other words, if N = (2)(3)(2)(5)(7)(2)(3), then we can be certain that N is divisible by 9.

So, if N = (2)(3)(2)(5)(7)(2)(3), then we can be certain that N is divisible by 2, 3, 4, 5, 6, 7, 8, and 9.
Similarly, if N = (2)(3)(2)(5)(7)(2)(3), then we can be certain that N is a MULTIPLE OF 2, 3, 4, 5, 6, 7, 8, and 9.

Does that help?