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A is sufficient because you know M is a multiple of 9. Since 9s is a multiple of 9 regardless of what s. You know N and M is consecutive multiples of 9 because their difference is 9. You know N is a multiple of 9 since if you add 9 to a number the only way it will form a multiple of 9 if it was already a multiple of 9.

Since you know their consecutive multiples of 9, their GCF is also 9.

B) Not enough info. You're not sure what number M is since, 4X+9 could be prime (13,17) or a multiple of 3 (21) or 5(25).

m = 4n + 9, where n is a positive integer. What is the GCD of m and n?

(1) m = 9s, where s is a positive integer --> since m is a multiple of 9 and is equal to 4n+9, then n must also be a multiple of 9 (in order 4n+9 to be a multiple of 9). Hence m and 4n are multiples of 9 and are 9 units apart from each other, which means that the Greatest Common Divisor of m and 4n is 9, obviously the GCD of m and n will also be 9. Sufficient.

USEFUL PROPERTY: if a and b are multiples of k and are k units apart from each other then k is greatest common divisor of a and b. For example if a and b are multiples of 7 and a=b+7 then 7 is GCD of a and b.

(2) n = 4t, where t is a positive integer. If n=4 then GCD of m and n is 1 (m=9 in this case) but if n=4*9 then GCD of m and n is 9 (m=17*9 in this case). Not sufficient.

A is sufficient because you know M is a multiple of 9. Since 9s is a multiple of 9 regardless of what s. You know N and M is consecutive multiples of 9 because their difference is 9. You know N is a multiple of 9 since if you add 9 to a number the only way it will form a multiple of 9 if it was already a multiple of 9.

Since you know their consecutive multiples of 9, their GCF is also 9.

B) Not enough info. You're not sure what number M is since, 4X+9 could be prime (13,17) or a multiple of 3 (21) or 5(25).

Therefore answer A is sufficient.

m cannot be 17 or 21 as in this case n won't be a multiple of 4. _________________

m = 4n + 9, where n is a positive integer. What is the GCD of m and n?

(1) m = 9s, where s is a positive integer --> since m is a multiple of 9 and is equal to 4n+9, then n must also be a multiple of 9 (in order 4n+9 to be a multiple of 9). Hence m and 4n are multiples of 9 and are 9 units apart from each other, which means that the Greatest Common Divisor of m and 4n is 9, obviously the GCD of m and n will also be 9. Sufficient.

USEFUL PROPERTY: if a and b are multiples of k and are k units apart from each other then k is greatest common divisor of a and b. For example if a and b are multiples of 7 and a=b+7 then 7 is GCD of a and b.

(2) n = 4t, where t is a positive integer. If n=4 then GCD of m and n is 1 (m=9 in this case) but if n=4*9 then GCD of m and n is 9 (m=17*9 in this case). Not sufficient.

Answer: A.

Hi Bunnel,

Had the question stem been like this > m=3n+9, then the GCF would hav been 3 right?

Re: m = 4n + 9, where n is a positive integer. What is the GCD [#permalink]
18 Aug 2014, 06:46

This is a problem from Manhattan advanced math.

m = 4n + 9

1) m = 9s 9s = 4n +9 n = (9s - 9) n = 9(s-1)/4 -> (s-1) must be multiple of 4 (n is integer) -> s could be 5, 9, 13, 17,... testing numbers for s, we can see that the GCD is always 9

suff

2) n = 4t testing: if t = 1, n = 4 and m = 25 -> GCD = 1 ( you could stop here if you are 100% sure about stat 1) if t = 2, n = 8 and m = 41 -> GCD = 1 if t = 3, n =12 and m = 57 -> GCD = 3

Not suff

gmatclubot

Re: m = 4n + 9, where n is a positive integer. What is the GCD
[#permalink]
18 Aug 2014, 06:46