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# What is the greatest common divisor of positive integers m

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What is the greatest common divisor of positive integers m [#permalink]

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19 Dec 2010, 23:27
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What is the greatest common divisor of positive integers m and n ?

(1) m is a prime number

(2) 2n = 7m
[Reveal] Spoiler: OA

Last edited by Bunuel on 02 Mar 2012, 12:22, edited 1 time in total.
Edited the question
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20 Dec 2010, 00:13
What is the greatest common divisor of positive integers m and n?

(1) m is a prime number --> if $$m=2=prime$$ and $$n=1$$ then $$GCD(m,n)=1$$ but if $$m=2=prime$$ and $$n=4$$ then $$GCD(m,n)=2$$. Two different answers, hence not sufficient.

(2) 2n=7m --> $$\frac{m}{n}=\frac{2}{7}$$ --> $$m$$ is a multiple of 2 and $$n$$ is a multiple of 7, but this is still not sufficient: if $$m=2$$ and $$n=7$$ then $$GCD(m,n)=1$$ (as both are primes) but if $$m=4$$ and $$n=14$$ then $$GCD(m,n)=2$$ (basically as $$\frac{m}{n}=\frac{2x}{7x}$$ then as 2 and 7 are primes then $$GCD(m, n)=x$$). Two different answers, hence not sufficient.

(1)+(2) Since from (1) $$m=prime$$ and from (2) $$\frac{m}{n}=\frac{2}{7}$$ then $$m=2=prime$$ and $$n=7$$, hence $$GCD(m,n)=1$$. Sufficient.

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20 Dec 2010, 00:24
From stmt 1 - since M is a prime number, and we do not have any info about n, we cannot say anything, hence insuff.
From stmt 2 - 2n = 7m. This statement does not say anything about m and n. It only says that m/n = 2/7 . The number could be anything {2,7} or {6, 21} . Both the cases produce different highest common divisor. So insuff.

Taking both the stmts together - what we know/deduce is - Divisors of product of two prime will be
1, the prime number1, the prime number 2, and the product of two prime num.
so for 7m = {1, 7, m , 7m}
and for 2n = 7m, given m to be a prime number, m has to be 2. If m is 2, then n = 7 and hence suff.

Hope it clears.
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Which is the greatest common divisor the two positive [#permalink]

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27 Jan 2013, 03:18
Hi everybody!

I have some difficulties with this question.

Which is the greatest common divisor the two positive integers m and n?

1) m is a prime
2) 2n=7m

Thanks!

Last edited by MacFauz on 27 Jan 2013, 05:02, edited 1 time in total.
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Re: Which is the greatest common divisor the two positive [#permalink]

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27 Jan 2013, 05:09
mjg2110 wrote:
Hi everybody!

I have some difficulties with this question.

Which is the greatest common divisor the two positive integers m and n?

1) m is a prime
2) 2n=7m

Thanks!

1) m can take several values and there is no information about n. Insufficient.
eg : m = 2, n = 1, GCF = 1
m = 2, n = 2, GCF = 2

2) m and n can take several values. Insufficient.
eg : m = 2, n = 7, GCF = 1
m = 4, n = 14, GCF = 2

1 & 2 together,

$$m = \frac{2}{7}n$$. So, n can only be 7 because any other value of n will either give a fraction or a non prime value for m.

Hence values of n & m are known. Sufficient.
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Re: Which is the greatest common divisor the two positive [#permalink]

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27 Jan 2013, 05:43
mjg2110 wrote:
Hi everybody!

I have some difficulties with this question.

Which is the greatest common divisor the two positive integers m and n?

1) m is a prime
2) 2n=7m

Thanks!

Merging similar topics. Please refer to the solutions above.
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Re: What is the greatest common divisor of positive integers m [#permalink]

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31 Jan 2013, 02:31
We have n and m as positive integers.

From F.S 1, we have m is a prime.Let us assume m=7. Thus, for n=14, we have gcd(m,n) as 7, for n=6 we have gcd(m,n) as 1. Thus this statement by itself is not sufficient.

From F.S 2, we have 2n=7m. Thus, n = 7m/2. Now as n,m are integers, m=2k(k is an integer). Thus, we get

n=7k, m=2k. As both 2 and 7 are prime, the gcd(m,n) here will be k, and this can have any value(1,2,3...);Thus not sufficient.

Combining both the F.S, we know m is prime and m=2k. Thus k can not be anything except 1, else m won't be a prime anymore. Thus, k=1 and n=7,m=2.

gcd(m,n) = gcd(2,7) = 1.

C.
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Re: What is the greatest common divisor of positive integers m [#permalink]

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15 Feb 2013, 18:44
jullysabat wrote:
What is the greatest common divisor of positive integers m and n ?

(1) m is a prime number

(2) 2n = 7m

What is the GCF of m & n?

(1) Insufficient- it can be any set of #'s (2 & 6, 3 & 12)
(2) Insufficient- you can plug in any #'s that make the equation equal (n=35 & m=10 - GCF is 5 or n=26 & m=8 - GCF is 2)

You know m is prime so the only way to balance out the equation is to replace m as 2 (only even prime #) b/c whatever 2n produces it will be an even #.

odd * odd = odd
even * even = even
even * odd = even
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Re: What is the greatest common divisor of positive integers m [#permalink]

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10 Jun 2016, 09:48
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Re: What is the greatest common divisor of positive integers m   [#permalink] 10 Jun 2016, 09:48
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