Why$ 1/12$ is NOT an irreducible basic fraction?

Question

I'm trying to solve this problem.

• A fraction $m/n$ is basic if $0 \le m < n$,
• It is irreducible if $\gcd( m,n ) = 1$ (greatest common divisor)

In the example, when $n=12$, irreducible basic fractions are: $0/12, 5/12, 7/12, 11/12$.

Why is $1/12$ not an irreducible basic fraction?

• $0 \le 1 < 12$
• $\gcd( 1, 12 ) = 1$.

Answer 1

@Vladislav It is an irreducible basic fraction. You're right. The source of the problem is wrong. It certainly matches both the criteria, so you're correct.

Edit: As stated in the comments, it looks like the source meant to say 1/12, but typed out 0/12...

Answer 2

The greatest divisor of $1$ is $1$. Thus the greatest common divisor of $1$ and any other number is $1$ (Since $1$ is a divisor of any other number).

Answer 3

It is. Your cited source is wrong. As it looks like a programming challenge, you might guess that they actually implemented the checker correctly, or perhaps they are always one off.

This is the difference between the answer being $\varphi(n)$ and $\varphi(n)+1$, where $\varphi$ is the Euler totient function.