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

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

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

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

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

• I'm not sure I understand the question. Is the question that $17\equiv 24\mod m$, and you need to find out $m$? – Guy Apr 29 '17 at 14:55
• $17 \in \bar{a}$ – Felipe Maia Apr 29 '17 at 15:02
• @FelipeMaia What is $\bar{a}$? – kccu Apr 29 '17 at 15:04
• @kccu Set consisting of all integers that are congruent to the integer $a\; mod\; m$ – Felipe Maia Apr 29 '17 at 18:33

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 $7$.
Hint: you have $24\equiv 17\pmod{m}$ if and only if $m\mid(24-17)$.
• So who is $m$? – Felipe Maia Apr 29 '17 at 19:34
• I understood that $24 = m \cdot k + r$. For this reason, the question "who is $m$?" – Felipe Maia Apr 29 '17 at 19:39
• @FelipeMaia What are the divisors of $24-17=7$? The formula “$a\mid b$” is read “$a$ is a divisor of $b$”. – egreg Apr 29 '17 at 20:07
• You did not understand my speech. I understand the concept of divisor, otherwise it would not have said "I thought of calculating m for the values ​​of the divisors of $24$"... Anyway, what I meant is that in this situation $24=m \cdot k + r$, how to get $m$?! – Felipe Maia Apr 29 '17 at 23:38
• @FelipeMaia You can't, in general. The problem at hand is not to compute $r$, but you have the information that $24$ and $17$ are congruent modulo $m$. – egreg Apr 30 '17 at 8:31