Because it doesn't :) The author is trying to use Euler's theorem (if a, n are coprimes, then a^{\phi(n)} \equiv 1 \mod n), but the map defined by the exponent

Because it doesn't :) The author is trying to use Euler's theorem (if a, n are coprimes, then a^{\phi(n)} \equiv 1 \mod n), but the map defined by the exponentiation doesn't say anything about what happens when (a, n) are not coprime. I'd encourage you to read this post, which is linked by the article: https://blog.antfeedr.com/posts/fizzbuzz.html https://blog.antfeedr.com/posts/fizzbuzz.html.