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.
_________________

GCD? Do you mean Lowest Common Multiple or Greatest Common Factor?

I think he means GCF.

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.
_________________

Hi Bunnel,

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

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.
(2) n = 4t, where t is a positive integer.

Bunnel : Interesting way of Defining the GCD of two numbers.

My thought process :
1)m=9s but m=4n+9. -> n is definately a multiple of 9. Hence GCD of m and n is 9. Sufficient.

2)n=4t. Doesn't help to establish any relation b/w m and n more than whatever is already mentioned. Insufficient.

Hence A

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

Why is it obvious that just because GCD of m and 4n is 9 therefore GCD of m and n will also be 9?

If the greatest common divisor of m and n were more that 9, say 18, then it would mean then 18 is a factor of both m and n, so 18 would also be a factor of m and 4*n and the GCD of m and 4n would not be 9.
_________________

GCD = Greatest common divisor
The greatest common divisor is also known as the greatest common factor (GCF), highest common factor (HCF), greatest common measure (GCM), or highest common divisor.

GMAC prefers the term GCD


_________________