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m=4n+9, where n>0 and is an integer. What is the GCF of

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m=4n+9, where n>0 and is an integer. What is the GCF of

by RBBmba@2014 » Mon Mar 23, 2015 7:18 am
m=4n+9, where n>0 and is an integer. What is the GCF of m and n ?

1. m=9s where s>0 and is an integer

2. n=4t where t>0 and is an integer

OA: A

P.S: I think,it can be solved by plugging in values and observing the pattern. However,is there any smarter/faster way to approach this type of problems ?
@Experts - could you please clarify and share your analysis ?
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by [email protected] » Mon Mar 23, 2015 11:28 am
HI RBBmba@2014,

This is a 'layered' DS question. While it has some subtle Number Properties built into it, for most Test Takers the best approach would be to TEST VALUES and look for a pattern. This is NOT a question that you're expected to answer in under a minute.

We're told that M = 4N+9 and that N is a POSITIVE INTEGER. We're asked for the greatest common factor of M and N

Before we get to the two Facts, there are some Number Properties worth noting. Since N is a positive integer and M = 4N+9, we know that M will be a positive integer. Furthermore, since 4N will always be EVEN, 4N+9 will always be ODD. Thus, M will always be a positive ODD integer.

Fact 1: M = 9S and S is a positive integer

From this, we know that M is a positive multiple of 9. We also know from the prompt that it's M is ODD. With a few TESTs, you can prove that a pattern exists and that the GCF is 9 (although it will take a little work to figure out the values of M that "fit").

You can also prove this through Number Properties:

Since M is a multiple of 9.....4N+9 is a multiple of 9

9 is a multiple of 9.....so for 4N+9 to be a multiple of 9.....4N must ALSO be a multiple of 9

Since 4N is also EVEN, we now know that 4N is an EVEN multiple of 9 and M is an ODD multiple of 9. The GCF of these two values MUST be 9.
Fact 1 is SUFFICIENT

Fact 2: N = 4T and T is a positive integer

IF....
T = 1
N = 4
M = 25
GCF = 1

T = 3
N = 12
M = 57
GCF = 3
Fact 2 is INSUFFICIENT

Final Answer: A

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by GMATGuruNY » Mon Mar 23, 2015 12:51 pm
I posted a solution for a very similar problem here:

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by Matt@VeritasPrep » Mon Mar 23, 2015 3:39 pm
Here's a nice property we can work with: for any positive integers x and y, the GCF of x and y MUST be a factor of (x - y).

So the GCF of m and n MUST be a factor of (m - n).

m = 4n + 9, so
m - n = 3n + 9

Now remember that the GCF, whatever it is, is a factor of n. Since it's a factor of n AND a factor of 3n + 9, we can use this property again.

GCF(m,n) is a factor of (3n + 9) - n, or a factor of 2n + 9

and again

GCF(m,n) is a factor of (2n + 9) - n, or a factor of n + 9

and again

GCF(m,n) is a factor of (n + 9) - n, or a factor of 9

So the GCF, whatever it is, is a factor of 9. That gives us three options for the GCF: 1, 3, and 9.

S1::

m = 9s

This tells us that m is a multiple of 9. We also know that

m = 4n + 9, so
9s = 4n + 9, so
9s - 9 = 4n, so n is a multiple of 9

Since m and n are both multiples of 9, 9 is a common factor of m and n. Since 9 is also the greatest choice from our original list (the GCF must be 1, 3, or 9), we KNOW 9 is the GCF, and we're done.

S2::

n = 4t, so

m = 4(4t) + 9
m = 16t + 9

This isn't terribly helpful. For instance, if t = 1, then m = 25, and the GCF of m and n = 1. But if t = 9, then the GCF of m and n is clearly 9.
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