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Mathematics

Coprime Numbers and Euler's Totient Function in Modular Arithmetic

Quick fact

For any number n, exactly φ(n) integers from 1 to n are coprime to n—and these are precisely the numbers that have a multiplicative inverse modulo n. For example, φ(10)=4 because only 1, 3, 7, and 9 are coprime to 10; indeed, each of these has an inverse modulo 10, such as 3·7 ≡ 1 (mod 10).

Why this is interesting

Think of the numbers from 1 to 10. Some of them can be multiplied together to get back to 1, but only if they share no secret factor. What's the hidden rule that decides which numbers are 'lucky'?