Math Homework: GCDs

Suppose that K is an integer between 600000 and 800000 such that gcd(k,271)=gcd(k, 2010)+70.
Find K.

I know the answer should be K=708123, but what are the steps I need to show for this question?

 

271 is a prime number, so its only 2 factors are 271 and 1 (you can test this by a few division tests of numbers like 17 and 13. Knowing this, we know that the gcd we are looking for is 271, since we are looking for a number greater than 70. Knowing this, we know that the gcd of (k, 2010) is 201. We now know that our number has 201 and 271 as factors. 201 and 271 are relatively prime, so we know that our number must be divisible by 201*271, which is 54471.

We also know that our number cannot be even, because if it were, then the GCD of it and 2010 would be 402, not 201. The only numbers divisible by 54471 and that are between 600k and 800k are 653652, 708123, and 762594. 708123 must be the answer, as it is the only one that isn't even

 

Hey, thanks a lot dude!

/sex