Having 2 numbers that guaranteedly have a point where they can be used to get any number, how can this point be estimated?

Edit: For example, I have the numbers 3 and 5 I know that I can get any number starting from 8 by adding diffrent combinations of these numbers ie 9 = 3+3+3 and 10 = 5+5. How, knowing that the numbers guaranteedly have this point (where I can get any number starting from x), how could I estimate what x is?

(repost because I cant unsolve a problem)

whatismath Jul 15, 2020

#1**+1 **

If I've understood the question correctly (and I might well not have!), it can be re-phrased as follows:

Assume I have two arbitrarily chosen integers, n and m.

Find another integer x such that for *any* integer, p, say, greater than or equal to x, that integer can be expressed as a sum of multiples of n and m. i.e. p = u*n + v*m, where u and v are positive integers (or one of them is zero).

If my interpretation is correct, then

(a) if n and m are both even, there is no such x, since one can never find values of u and v such that u*n + v*m = p when p is an odd number.

(b) If one or both of n and m is/are odd, then I speculate that x = n*m (but I've no proof, and I haven't tried hard to find a counter-example!).

Alan Jul 15, 2020