The integer $17$ belongs to the residue class modulo $m$ of $24$. Find $m$.

Question

Please, help me on this question. I decided, but I need to know if my answer is correct.

I thought of calculating m for the values ​​of the divisors of 24, that is, making m belonging to D (24).

So I found

$m=2 \Rightarrow \{2k+1;k \in \mathbb{Z}\}$

$m=3 \Rightarrow \{3k+2;k \in \mathbb{Z}\}$

$m=4 \Rightarrow \{4k+1;k \in \mathbb{Z}\}$

Answer 1

Definition of residue. The number $r$ in the congruence $a\equiv r\pmod m$ is called the residue of $a\pmod m$. In the case at hand $r=17$ and $ a=24 $.

This means that for some integer $k$ the following equality holds $24=17+km $. You should then have $km=7$, where $ k $ and $ m $ are positive integers. This implies that $ m=7 $, because $7 $ is a prime number, that is, it has no divisors, except $ 1 $ and $7 $.

Answer 2

Hint: you have $24\equiv 17\pmod{m}$ if and only if $m\mid(24-17)$.