Aide

Suppose you have an infinte supply of coins of value p, and infinite supply of coins of value q. You want to give money away.
which amounts can you give?

what the values of m.p+n.q, where m and n are natural integer numbers, could be?
In fact, m.p+n.q = (m.p' +n.q')*gcd(p,q), where gcd(p,q) indicates the greater common dividor of p and q, and p' and q' the respective ratios of p and q to gcd(p,q), such that gcd(p',q') =1. Therefore, could (m.p'+n.q') have any natural value? No. Indeed, its smaller non null values, if supposed that q' < p' for example, are q', 2q', 3q'..., [p'/q']q', p', ([p'/q']+1)q', p'+q', ([p'/q']+2)q', p'+2q', ([p'/q']+3)q', p'+3q',..., 2[p'/q']q', p'+[p'/q']q', 2p' and (2[p'/q']+1)q', p'+([p'/q']+1)q', 2p'+q' and (2[p'/q']+2)q'..., where [x] symbolizes the integer part of x. Morover, if m.p'+n.q' = y.p'+z.q', (m,n) and (y,z) being natural integers pairs, then (m-y)p' = (-n+z)q', and because gcd(p',q') =1, it exists an integer a such that z-n =a.p', and m-y =a.q', so z=n+a.p' and y = m-a.q'. Then, with z and y non negative, -[n/p'] =< a =< [m/q']. Thus for each (m.p'+n.q') there are ([m/q']+[n/p']) others pairs (m,n) which give the same value.