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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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Bunuel wrote:
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.

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 ?
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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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