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# What is the greatest common divisor of positive integers m

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

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22 Dec 2017, 11:54
Top Contributor
lahoosaher wrote:
What is the greatest common divisor of positive integers m and n.

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

Target question: What is the GCD of m and n?

Statement 1: m is a prime number
If m is a prime number, it has exactly 2 divisors (1 and m), so this tells us that the GCD of m and n must be either 1 or m.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT.

Statement 2: 2n = 7m
If 2n = 7m then we can rearrange the equation to get n = (7/2)m

IMPORTANT: Notice that if m were to equal an ODD number, then n would not be an integer. For example, if m = 3, then n = 21/2 (n is not an integer). Similarly, if m = 11, then n = 77/2 (n is not an integer). So, in order for n to be an INTEGER, m must be EVEN.

If m must be EVEN, there are several possible values for m and n. Consider these two cases:
case a: m = 2 and n = 7, in which case the GCD = 1
case b: m = 4 and n = 14, in which case the GCD=2
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT.

Statements 1 & 2 combined
From statement 1, we know that m is prime, and from statement 2, we know that m is even.
Since 2 is the only even prime number, we can conclude that m must equal 2.
If m = 2, then n must equal 7, which means that the GCD must be 1.
Since we are able to answer the target question with certainty, statements 1 & 2 combined are sufficient, and the answer is C

Cheers,
Brent
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Re: What is the greatest common divisor of positive integers m  [#permalink]

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21 Oct 2018, 02:08
(1) m is primer. Insuff
(2) m/n=7/2. m is multiple of 7, n is multiple of 2. GCD=1 but if, 24 and 4 than GCD=4. Insuff

(1)(2) m is prime and m is multiple of 7 So, m is 7
if m is 7, then n is 2
GCD=1
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Re: What is the greatest common divisor of positive integers m  [#permalink]

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23 Nov 2018, 10:13
Bunuel wrote:
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.

2) 2n=7m --> m/n=2/7

n=3.5m

GCF(m,3.5m)= m ? is this correct ? Its not sufficient because we dont have the value of M ?
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Re: What is the greatest common divisor of positive integers m  [#permalink]

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23 Nov 2018, 21:55
renjana wrote:
Bunuel wrote:
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.

2) 2n=7m --> m/n=2/7

n=3.5m

GCF(m,3.5m)= m ? is this correct ? Its not sufficient because we dont have the value of M ?

Hello

Yes, I think you have concluded properly. GCF of m & 3.5m will depend on the value of m. Eg, if m= 2, then 3.5m = 7, and their GCF will be 1.
However, if m= 4, then 3.5m= 14, and their GCF will be 2. So GCF can take multiple values depending on the value of m.
Re: What is the greatest common divisor of positive integers m   [#permalink] 23 Nov 2018, 21:55

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