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

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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

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Lahoosaher

Originally posted by lahoosaher on 07 Jun 2009, 10:12.
Last edited by Bunuel on 06 Jun 2019, 07:23, edited 3 times in total.
Edited the question and added the OA
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Re: What is the greatest common divisor of positive integers m and n ?  [#permalink]

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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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What is the greatest common divisor of positive integers M and N ???
1)M is a prime number
2)2N=7M

If we look at statement 2 and plug in numbers we'll quickly see it's not sufficient.

Let M=2 then N = 7 GCD=1.
Let M=6 then N = 21 GCD=3.

S2 basically tells us that 2 is a factor of M and 7 is a factor of N. But we don't know if they have more shared factors or not.
Insufficient.

Does that make sense Akshaydiljit?

Originally posted by testprepDublin on 04 Jul 2011, 07:53.
Last edited by testprepDublin on 05 Jul 2011, 01:14, edited 1 time in total.
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Re: What is the greatest common divisor of positive integers m and n ?  [#permalink]

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2
OA is C.

1)m is prime
Clearly insufficient.

2)2n=7m
can be written as n= 7m/2. n & m are integers.
Put m=1,2,3,4 .... therefore m has to be a multiple of 2.
Insufficient.

Combined-
m is prime(stat1) and m= multiple of 2(stat2)

Hence m=2 & n=7

GCF is 1.
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Re: What is the greatest common divisor of positive integers m and n ?  [#permalink]

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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.

Greatest Common divisor and Highest common factor are same thing Bunuel?

Because n= 7m/2 (Taking both this is true only for m = 2) So Greatest common divisor is 2 not 1, Isn't it?
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Re: What is the greatest common divisor of positive integers m and n ?  [#permalink]

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honchos 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.

Greatest Common divisor and Highest common factor are same thing Bunuel?

Because n= 7m/2 (Taking both this is true only for m = 2) So Greatest common divisor is 2 not 1, Isn't it?

Yes, GCD and GCF are the same thing.

But couldn't understand your second point: the greatest common divisor of 2 and 7 is 1. How can it be 2? Is 7 divisible by 2?
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Re: What is the greatest common divisor of positive integers m and n ?  [#permalink]

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Bunuel,
Our m is coming as 2, so isn't 2 a GCD, Or may be I have misunderstood the solution?
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Re: What is the greatest common divisor of positive integers m and n ?  [#permalink]

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honchos wrote:
Bunuel,
Our m is coming as 2, so isn't 2 a GCD, Or may be I have misunderstood the solution?

The question asks: what is the greatest common divisor of positive integers m and n?

We got that m=2 and n=7. What is the greatest common divisor of 2 and 7? Is it 2? No, it's 1.
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Re: What is the greatest common divisor of positive integers m and n ?  [#permalink]

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lahoosaher wrote:
What is the greatest common divisor of positive integers m and n.

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

We need to determine the greatest common divisor, or the greatest common factor (GCF), of integers m and n.

Statement One Alone:

m is a prime number.

Since we don’t know anything about n, statement one is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

2n = 7m

We can manipulate the equation 2n = 7m:

n = 7m/2

n/m = 7/2

Even with the equation rewritten, we see that there are many options for m and n, and thus there are many different GCFs for m and n. For instance, if n = 7 and m = 2, then the GCF is 1. However, if n = 14 and m = 4, then the GCF is 2. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Using statements one and two, we know that m is prime and that n/m = 7/2. Therefore, m must equal 2 and n must equal 7. When m is 2 and n is 7, the GCF is 1.

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

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

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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 and n ?  [#permalink]

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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.
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