It's just modular math. The output is fizz if the output is orthogonal to 0 mod 3, and buzz if it is orthogonal to 0 mod 5. It's fizzbuzz when it's 0 mod 15. N

It's just modular math. The output is fizz if the output is orthogonal to 0 mod 3, and buzz if it is orthogonal to 0 mod 5. It's fizzbuzz when it's 0 mod 15. Now, since 15 is divisible by 3, it's easy to see that x = 0 mod 3 iff x = {0, 3, 6, 12} mod 15, and similarly x = 0 mod 5 iff x = {0, 5, 10} mod 15. So you can see now that you can determine the output for a given index based on the remainder after dividing by 15, so the sequence must repeat every 15 elements.