Abstract Algebra - Natural Numbers Proof

adamsmc2

The question is which sets of natural numbers are closed under addition. I know that odd is not, and I know how to prove that sets of multiples are, but my professor said there is something more and that is has to do with greatest common divisor. He said to pick numbers like 3 and 5 or 5 and 8, then expand the set.

For example 3,5 would be {3,5,6,8,9,10,11,12,13,14,15,....}

He said we should be able to observe over multiple sets something that is not completely obvious but I can't see anything. Possibly it has something to also do with the Euclidian Algorithm but I'm not so sure about that. Also, he said something about when the set starts to show regularity.

Any insight will help. Thanks!

CompuChip

I think you are on the right track. Note that you happened to choose two co-prime numbers. Do you see how at some point, you start to get all numbers (10, 11, 12, ...)?
Given 3 and 5, if you perform the Euclidean Algorithm, that basically gives you ... what?

lurflurf

Let n be any natural number

If a is in the set a*n must be
If a and b are in the set, gcd(a,b)*n is, if n is large enough.