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

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m = 4n + 9, where n is a positive integer. What is the GCD [#permalink]  14 Feb 2012, 04:16
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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.
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Re: Integers [#permalink]  14 Feb 2012, 04:55
Smita04 wrote:
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.

GCD? Do you mean Lowest Common Multiple or Greatest Common Factor?
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Re: Integers [#permalink]  14 Feb 2012, 05:05
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).

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Re: Integers [#permalink]  14 Feb 2012, 05:24
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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.

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Re: Integers [#permalink]  14 Feb 2012, 05:27
Expert's post
kys123 wrote:
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).

m cannot be 17 or 21 as in this case n won't be a multiple of 4.
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Re: Integers [#permalink]  05 May 2013, 08:23
Bunuel wrote:
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.

Hi Bunnel,

Had the question stem been like this > m=3n+9, then the GCF would hav been 3 right?
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Re: m = 4n + 9, where n is a positive integer. What is the GCD [#permalink]  05 May 2013, 10:12
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
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If m = 4n + 9, where n is a positive integer. What is the gr [#permalink]  05 Sep 2013, 21:56
If m = 4n + 9, where n is a positive integer. What is the greatest common factor of m and n?

(1) m = 9s, where s is a positive integer.
(2) n = 4t, where t is a positive integer.
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Re: GCD of m & n, where m = 4n + 9 [#permalink]  05 Sep 2013, 22:05

m-4n-9-where-n-is-a-positive-integer-what-is-the-gcd-127523.html
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Re: If m = 4n + 9, where n is a positive integer. What is the gr [#permalink]  05 Sep 2013, 22:15
Expert's post
AkshayChittoria wrote:
If m = 4n + 9, where n is a positive integer. What is the greatest common factor of m and n?

(1) m = 9s, where s is a positive integer.
(2) n = 4t, where t is a positive integer.

Merging topics. Please refer to the solutions above.
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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
Re: m = 4n + 9, where n is a positive integer. What is the GCD   [#permalink] 18 Aug 2014, 06:46
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