Vincen wrote: ↑

Wed Sep 15, 2021 8:52 am

\(M\) and \(N\) are integers such that \(6<M<N.\) What is the value of \(N?\)

(1) The greatest common divisor of \(M\) and \(N\) is \(6.\)
(2) The least common multiple of \(M\) and \(N\) is \(36.\)

Answer: C

Source: Official Guide

Given: M and N are integers such that 6 < M < N

Target question: What is the value of N?

Statement 1: The greatest common divisor of M and N is 6
There are infinitely many pairs of values that satisfy the statement 1.
Here are two cases:
Case a: M = 12 and N = 18. In this case, the answer to the target question is N = 18
Case b: M = 18 and N = 24. In this case, the answer to the target question is N = 24
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The least common multiple of M and N is 36
There are at least two pairs of values that satisfy the statement 2:
Case a: M = 12 and N = 18. In this case, the answer to the target question is N = 18
Case b: M = 18 and N = 36. In this case, the answer to the target question is N = 36
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that the greatest common divisor of M and N is 6
Statement 2 tells us that the least common multiple of M and N is 36

At this point, there are at least two different approaches we can take...

APPROACH #1: List possible pairs of values that satisfy both statements
The good thing here is that statement 2 indirectly tells us N is less than or equal to 36
We know this because statement 2 tells us that 36 is a multiple of N.
So, let's first list all pairs of values that satisfy statement 2 (as well as the given information):
i) M = 12 and N = 36
ii) M = 18 and N = 36
iii) M = 12 and N = 18
That's it!!

Among these three possible pairs of values, only one pair satisfies statement 1: M = 12 and N = 18
Since it must be the case that M = 12 and N = 18, the target question is N = 18


APPROACH #2: Apply a useful rule
--------ASIDE----------------------
There's a nice rule that says:
(greatest common divisor of x and y)(least common multiple of x and y) = xy
Example: x = 10 and y = 15
Greatest common divisor of 10 and 15 = 5
Least common multiple of 10 and 15 = 30
Notice that these values satisfy the above rule, since (5)(30) = (10)(15)
--------BACK TO THE QUESTION! ----------------------
When we apply the above rule, we see that MN = (6)(36) = 216
In other words, MN = (2)(2)(2)(3)(3)(3)
Since it must be true that 6 < M < N ≤ 36, we can see that our options are very limited.
For example it COULD be the case that M = (2)(2)(2) = 8 and N = (3)(3)(3) = 27, but this pair of values does not satisfy statement 1 or statement 2

It COULD also be the case that M = (2)(2)(3) = 12 and N = (2)(3)(3) = 18, AND this pair of values does satisfy statements 1 and 2
Is there any other pairs of values we can use so that 6 < M < N ≤ 36 and MN = 216?
The answer is no.
So, it must be the case that M = 12 and N = 18, the target question is N = 18

Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent