"Why do we have 1p and 2p coins? There's nothing that only costs a penny or tuppence these days!"
"We need them, to make sure we can make all of the bigger amounts!"
"Well couldn't we do that with some bigger coins?"
"If we only had 3p and 7p coins, there'd be some values that we couldn't make, like 4p or 8p!"
"Yes, but I reckon we'd be able to make most amounts, even if we only had 3p and 7p coins..."
Which values is it impossible to make with 3p and 7p coins?
Given an amount, is there a quick way to work out how to make it?
Given an amount, is there a quick way to work out how to make it with the least number of coins?
You might like to try Frobenius Numbers next.