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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m=4n+9, where n>0 and is an integer. What is the GCF of
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RBBmba@2014
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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
GMAT assassins aren't born, they're made,
Rich
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
GMAT assassins aren't born, they're made,
Rich
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I posted a solution for a very similar problem here:
https://www.beatthegmat.com/greatest-com ... 10738.html
https://www.beatthegmat.com/greatest-com ... 10738.html
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
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As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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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.
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.
