The Euler phi function counts the number of elements in the range 1 to n which are coprime to n. This is the same as the number of invertible elements modulo n.

The sketch below demonstrates the relationship between phi(a), phi(b) and phi(ab). Use the keys “ijkl” like arrow keys to resize the grid. To understand the cascading integers, see my sketch on the Chinese Remainder Theorem. The sketch is a proof without words that phi(ab) = phi(a)*phi(b) whenever a and b are coprime. (You can also use the sketch to discover the formula for phi in general, with enough patience.)