If $n=\frac{a}{b}$ , with $n, a$ and $b$ integers $( b ≠ 0)$, and with $a$ and $b$ coprimes, does $b=1$?

Question

If $n = { \dfrac{a}{b} }$ , with $n$ ${ \in }{ \mathbb{N} }$, $a$ ${ \in }$ ${ \mathbb{N} }$, and $b$ ${ \in }$ ${ \mathbb{N^*} }$ , and if $a$ and $b$ are coprime, then does $b$ $=$ $1$ ?

I can only see that as an option, but I'm not enirely sure. A little help ?

Thanks.

$bn=a$ then $b$ is a factor of $a$. As $(a,b) =1$ then $b=1$ — , Aug 29 '20 at 17:00
Immediate consequence of the linked uniqueness of reduced fractions, i.e. $,\dfrac{a}b = \dfrac{n}1,\Rightarrow, b = 1\ $ — Bill Dubuque, Aug 29 '20 at 17:42

score 3 · Answer 1 · edited Aug 29 '20 at 18:29

3

Yes, since $b\mid a$ and $b\mid b$ hence $b|\gcd(a,b)=1$, hence $b=1$.

edited Aug 29 '20 at 18:29

James Moriarty

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answered Aug 29 '20 at 17:01

Tabludif

159

If $n=\frac{a}{b}$ , with $n, a$ and $b$ integers $( b ≠ 0)$, and with $a$ and $b$ coprimes, does $b=1$?

1 Answers1