Inten21 wrote:

BunuelCan you please provide an elaborated Solution for this problem?

Would really appreciate it if you could explain in detail as to how to think and approach such tough LCM and GCF problems?

Also, can you link some more SIMILAR QUESTIONS if it is possible.

Thanks in advance.

While your question is to

Bunuel, I will try to share an detailed explanation that may help you as well as others.

What is the GCD - It is the product of the common terms with the least power (This is in my words and not a textbook perfect definition

)

E.g. If there are two integers 12 and 16

Step-1: Break into prime factors. 12 would be \(2^2*3\) and 16 would be \(2^4\)

Step-2: Identify common prime(s) i.e. 2

Step-3: Pick the common prime with the lowest power i.e. \(2^2\) in our example and that is your GCD

GCD - \(2^2\)

Conceptually GCD is the largest number that can divide the 2 integers in question. Try to find an integer greater than 4 that divides both 12 and 16. You won't be able to find one!

What is LCM - It is the product of the common terms with the highest power (Again this is in my words and not a textbook perfect definition

)

In the above example, pick out the highest power of all distinct primes i.e. \(2^4\) and \(3^1\)

Hence LCM would be \(2^4*3^1 = 48\)

Conceptually LCM is the smallest multiple of the 2 integers in the question. Try to find an integer less than 48 that is a multiple of both 12 and 16. You won't be able to find one!

On to the question at hand:-

Info. provided in the question:

1. Both M and N are integers

2. Both are greater than 6

3. N is the greater than M (\(N>M\), E.g. Least values of N could be 8 and that of M could be 7)

We are asked to determine the value of NStatement-1: GCD (Greatest Common Divisor) of M and N is 6 (or \(2*3\))

This states that 6 will be common to both M and N. Plus, as M and N are greater than 6 there will be other primes too. But this statement does not provide insight into those other primes and hence this statement is insufficient. Let me demonstrate that:

\(M = 2*3*5 = 30\)

\(N = 2*3*7 = 42\)

OR

\(M = 2*3*11 = 66\)

\(N = 2*3*13 = 78\)

Here GCD (M, N) is 6 but N can take different values while staying true to the 3 data points provided by the question stem.

Statement-2: LCM (Least Common Divisor) of M and N is 36 (or \(2^2*3^2\))

Basis the definition of LCM, in which we consider the highest powers of all distinct primes, this statement provides information that 2 and 3 are the only primes carried by M and N. But it does not provide insight into the powers of 2 and 3 specific to M and N. E.g. in the examples below \(2^2\) can be part of M in the first example and also part of N in the very last example, and hence this statement is insufficient.

\(M = 2*3*2 = 12\)

\(N = 2*3*3 = 18\)

\(M = 3*3 = 9\)

\(N = 2*3*2 = 12\)

Combining both statements:

When combining we need to ensure that:

1. \(N>M\) - This one is the key!

2. N and M are greater than 6

3. From statement-1 we know that 6 is common to both M and N

4. From statement-2 we know that 2 and 3 are the only primes carried by M and N and their highest powers are 2 (\(2^2\) and \(3^2\))

\(M = 2*3*2 = 12\)

\(N = 2*3*3 = 18\)

You cannot do the below as it would violate the condition that N>M (pt. 1 i.e. Key).

\(M = 2*3*3 = 18\)

\(N = 2*3*2 = 12\)

Ans C (or \(N = 18\))

While it sounds very simple, but as you can observe that the best approach to solving these questions and math questions in general is to list down the various possibilities in an organized fashion (one below another) in your notebook. The biggest mistakes happen when we ignore the data points provided in the question stem E.g. N>M or N,M>6 in this case.

Hope it helps!

_________________

Cheers. Wishing Luck to Every GMAT Aspirant!