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?

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

Your textbook is lying. 1/12 is indeed an irreducible fraction according to your definition — AgentS, Mar 18 '15 at 14:09
I think it's a typo and they meant 1/12 instead of 0/12. Gcd(0, 12) = 12 — Ruben, Mar 18 '15 at 14:09

score 2 · Accepted Answer · answered Mar 18 '15 at 14:10

2

@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...

answered Mar 18 '15 at 14:10

Panglossian Oporopolist

5,223

score 1 · Answer 2 · answered Mar 18 '15 at 14:09

1

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).

answered Mar 18 '15 at 14:09

Asinomás

105,651

score 1 · Answer 3 · answered Mar 18 '15 at 14:09

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.

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

3 Answers3