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Finding the Greatest Common Divisor is achieved by running through the factors of two or more numbers and find the largest number that they all have in common. For convenience sake lets say the two number we're comparing are $ m,n$ Would it be acceptable to define GCD as: $(\forall{m,n}\in \mathbb{Z})(\exists{x}\in\mathbb{Z}) ((x|m)\land(x|n))$

I don't know if this a correct definition for, but I feel like It's on the right track.

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    Your proposed formula has no free variables! How does it define anything? – 2017-01-24
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    The GCD of $m$ and $n$ is a number (that depends on $m$ and $n$). What you wrote down is not a number that depends on $m$ and $n$. You wrote down a proposition that says "For every pair of integers $m$ and $n$, there is an integer that divides both $m$ and $n$. That is a true fact, but it doesn't define the GCD. – 2017-01-24
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    Did you see [this question](http://math.stackexchange.com/questions/243083/gcd-using-quantifiers-where-did-i-make-a-mistke?rq=1), using quantifiers for the gcd? – 2017-01-24
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    $x=1$ as you have defined it above. Take $m,n$ are taken as givens. And $x$ not only has to divide both $m$ and $n$, it must be the largest integer to do so. – 2017-01-24
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    Your formula says that any integers m,n have a common divisor x. Which is true, but it is not the def'n of the largest such x. – 2017-01-24

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This is incomplete. You also have to state that no larger $x$ works.

For example, $x=1$ satisfies your conditions.

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    This isn't a terribly helpful answer. There's not really a simple way to add the additional piece of information you mention to what the original poster wrote and make it into a *definition* of the GCD. – 2017-01-24
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    Just add "and for all y > x either $y \not\mid n$ or $y \not\mid m$". – 2017-01-24
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    The g.c.d. is the greatest divisor for the partial order defined by the divisibility relation. – 2017-01-24
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    @martycohen See my comment to the accepted answer. A mathematical statement of existence (For all pairs of integers, there exists...) is not a definition. It might be a useful true statement, but it’s not a definition. – 2017-01-25