$n$ is divisor of $a-b$ if and only if $n$ is a divisor of $b-a$

Question

I am asked to show that $n$ is divisor of $a-b$ if and only if $n$ is a divisor of $b-a$

It may seem trivial but I am struggling with this question. I am trying to proof it by using the definition but I am struggling. The answer I have is that this question is trivial but I would really appreciate it if you guys could give me hand. Thanks.

What definitions are you aware of? – Aryaman Maithani Jul 01 '21 at 08:28 — Aryaman Maithani, Jul 01 '21 at 08:28

Sayan Dutta · Accepted Answer · 2021-07-01T13:35:14.657

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$(a-b)$ is divisible by $n$ if and if $a-b=kn$ for some integer $k$.

This implies $b-a=-(a-b)=(-k)\cdot n$. This means $(b-a)$ is divisible by $n$.

Does that help?

edited Jul 01 '21 at 13:35

answered Jul 01 '21 at 08:39

Sayan Dutta

8,831

OwO what's this? :3c – Tundra Shark Jul 01 '21 at 08:44
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@TundraFizz Anything wrong? – Sayan Dutta Jul 01 '21 at 08:45
Nope looks good to me – Tundra Shark Jul 01 '21 at 08:45
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i need to have more confidence in myself, that's exaclty what i wrote. thanks for your time – Gidon Avziz Jul 01 '21 at 09:04
Writing $b-a=-(a-b) = -(kn) = (-k)n$ would be clearer. – lhf Jul 01 '21 at 10:44
@Ihf That was a good suggection. Done the editing. – Sayan Dutta Jul 01 '21 at 13:35

$n$ is divisor of $a-b$ if and only if $n$ is a divisor of $b-a$

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