# Greatest Common Divisor Proof

1. Apr 2, 2012

### hoopsmax25

1. The problem statement, all variables and given/known data

Suppose a, b ∈ N and a|b. Prove that a = gcd(a, b).

2. Relevant equations
Seems easy intuitively but actually proving it is giving me problems.

3. The attempt at a solution
I have been trying to use the fact that gcd(a,b)=na + mb here m and n are integeres but got stuck.

2. Apr 2, 2012

### Poopsilon

Well we clearly have a|a and a|b, and the largest divisor of a number is itself. This should follow immediately.

3. Apr 2, 2012

### hoopsmax25

Yeah and i understand that but we are asked to prove it, not explain why.

4. Apr 2, 2012

### Poopsilon

Well you can probably prove by contradiction that the largest divisor of a number is itself. I mean a|b => a=gcd(a,b) is so trivial that there's very little to say, how formal does your professor expect your proof to be? You can take it all the way down to pure logic if you really wanted to.

5. Apr 2, 2012

### Poopsilon

The difference between a proof and an explanation actually becomes increasingly fuzzy the farther you get in math.

6. Apr 2, 2012

### micromass

Staff Emeritus
A proof of this is certainly possible.

But first, we would have to know how exactly you defined a|b and gcd(a,b).