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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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alphonsa wrote:
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

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

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