Frobenius coin problem

Statement

Let $a$ and $b$ be coprime integers with $a > 1$ and $b > 1$. Then $ab - a - b$ is not of the form $ax + by$ with natural numbers $x$ and $y$. Every integer $n > ab - a - b$ is of this form.

Remarks

The number $ab - a - b$ is the Frobenius number of $a$ and $b$: the largest amount that one cannot pay with coins of values $a$ and $b$. For example, with coins of $3$ and $5$, the amount $7$ is impossible, but $8$, $9$, $10$ and all larger amounts are possible. Exactly $(a - 1)(b - 1)/2$ positive integers are not of the form $ax + by$. For three or more coin values, no such simple formula is known.

History

James Joseph Sylvester found the formula for two coins in 1882.

Explanations

L1 human

With coins of values $a$ and $b$, you can pay all large amounts when $a$ and $b$ are coprime. The theorem gives the largest amount that you cannot pay: $ab - a - b$.