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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)

 Jul 15, 2020
 #1
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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!).

 Jul 15, 2020
edited by Alan  Jul 15, 2020
edited by Alan  Jul 15, 2020
edited by Alan  Jul 15, 2020
 #2
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Thanks for the answer

whatismath  Jul 15, 2020

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