\(M\) 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

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

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## Global Stats

**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