What is the greatest common divisor of positive integers m and n ?

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.

Answer: C.
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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

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?

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.

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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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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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.

Answer: C
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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.

Hi Bunuel - when you combine the statements, you mentioned that m = 2

How can you be sure that n = 7 always ? Why can't n = 14 for example (When m = 2)

Thank you

(2) says that 2n = 7m. If you substitute m = 2, there you'd get n = 7.
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Video solution from Quant Reasoning starts at 13:15
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Can you help me understand this more?

If using both statements , we plug in values in , 2n=7m .Then we will be arriving at different values? what to do in that case?

[/quote]

Can you help me understand this more?

If using both statements , we plug in values in , 2n=7m .Then we will be arriving at different values? what to do in that case?[/quote]

Could you clarify your question please? I also suggest watching my video solution above if you haven't already.

2n=7m tells us that 7m is Even
For 7m to be even, m has to be Even
We know that m is Prime : 2 is the only prime even number

So m=2
And n has to be 7

Posted from my mobile device
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Solution:

If there are no restrictions on n and m other than that both of them are positive (which is the case when we assume only statement two), then the equation 2n = 7m indeed has more than one solution (such as n = 21, m = 6 or n = 28, m = 8 or n = 35, m = 10). However, when we use both statements, we are told that m is a prime number. Furthermore, m must be even since 7m equals an even number (2n is even regardless of the value of n). Since m is even and prime, the only possible value for m is 2. Thus, the only possible value of n is 7.
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Statement 1:

Let's consider two scenarios:
(1) M = 2 and N = 3, then GCF = 1 (note: GCF of any two consecutive integers is equal to 1.)
(2) M = 2 and N = 4, then GCF = 2

So we've got two different answers to the GCF so not sufficient.

Statement 2:
If 2n=7m then that means that N = (7M)/2. Now remember that the stem provides us with the information that M and N are both integers. So the quotient of (7M)/2 MUST be an integer, and the only way that (7M)/2 can be an integer is if M is EVEN. Okay but again, this statement alone isn't sufficient, but lets just test two scenarios to confirm:

So now we need to find the GCF of (7M)/2 and M because N = (7M/2).
(1) M = 2 then GCF = 1
(2) M = 4 then GCF = 2

(1) and (2) together:
Well since M must be both a prime number, AND an even number then the only number that M can be is 2. So the GCF of (7M)/2 and M is gcf(7, 2) = 1.