Three Results on Making Change (An Exposition)

Assume you an infinite supply of pennies, nickels, dimes, and quarters (or some other finite set of denominations which are relatively prime). Let CH(n) be the number of ways to make change of n cents. We present a simple unified exposition of three know theorems about CH(n). Let M be the LCM of a1,...,aL. Let M' be the LCM of the GCD of all pairs of ai's. (1) If 0\le r\le M-1 then CH(n) restricted to n \equiv r mod M is a poly, (2) If 0\le r\le M'-1 then CH(n) restricted to n\equiv r mod M' is a poly except for the constant term, (3) CH(n) is n^{L-1}/(L-1)!a1a2...aL + O(n^{L-2}). Part (3) is known as Schur's theorem.