What is the greatest common divisor of positive integers m

Question

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

(1) m-n and n are co-prime
(2) m and n are consecutive integers

Source: 4Gmat

Bunuel · Accepted Answer

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

(1) m-n and n are co-prime. This means that the greatest common divisor of m-n and n is 1. Now, if m and n had greatest common divisor greater than 1, then m-n would also share the same factor (if x is a factor of both m and n, then x must also be a factor of m-n), thus in this case m-n and n would have that factor as the greatest divisor not 1. Therefore m and n do not share any factor greater than 1. Sufficient.

(2) m and n are consecutive integers. Consecutive integers are co-prime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). Thus the greatest common divisor of m and n is 1. Sufficient.

Answer: D.

alphonsa · Answer

When you explain it, it seems quite easy

, but while working it out, it doesn't seem to be that way. One reason is because, I don't necessarily think in that same line

Any suggestions ? :cry:

chetan2u · Answer

Hi,

(1) m-n and n are co-prime 
what ever common factors m and n have , m-n, m+n, m and n will also have same factors..

Reason  - say m and n have x in common....  so  m = xa  and  n = xb. ...
so  m-n = xa-xb = x(a-b)..................m+n = x(a+b)................m=xa  .............n=xb......mn = xa*xb...........
BUT not \frac{m}{n} 
thus all five of them have same common factors..

so IF m-n and n are co-primes or both do not have any factor in common, so m and n will also be co-prime..
Suff

(2) m and n are consecutive integers 
consecutive integers do not have any factor other than 1 in common.....
Suff

D

Pritishd · Answer

Here's another approach to solving statement one, though I am not totally sure if this is the correct approach.

We know that co-primes do not share any common factors and from a pattern point of view if two integers are separated by  1  unit (consecutive integers) then we can be sure that the two integers are co-primes E.g.  k  and  k + 1  were  k  is a positive integer

Now  n  and  n + 1  will be co-primes for sure and we are told that  m - n  and  n  are co-primes, so we could write  n + 1 = m - n  which upon simplification yields  m = 2n + 1

When,
 n = 1 ,  m = 3  and GCD  = 1 
 n = 2 ,  m = 5  and GCD  = 1 
 n = 3 ,  m = 7  and GCD  = 1 
 n = 4 ,  m = 9  and GCD  = 1 
 n = 15 ,  m = 31  and GCD  = 1 
 n = 20 ,  m = 41  and GCD  = 1 
and so on...

Again I am not sure if this is the right approach. I just shared what's running in my mind!

What is the greatest common divisor of positive integers m

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What is the greatest common divisor of positive integers m

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