Why can we say that if $d$ divides both $a$ and $b$, then it also divides the remainder of $a/b$?

Question

For example if $a=121$ and $b=44$, then we could say that $d=11$. The remainder of $121/44$ is $33$, and this is also divisible by $11$.

score 3 · Accepted Answer · answered Aug 01 '19 at 17:08

3

By the Division Algorithm, we can write $a=bq+r$ for unique integers $q$ and $r$. In this case, $r$ is called the remainder.

Now, if $d$ divides $a$ and $b$, it also divides any linear combination of these, in particular $$d\mid a-bq =r$$

answered Aug 01 '19 at 17:08

azif00

20,792

1

$r$ is unique under the constraint $0 ≤ r < b$ – J. W. Tanner Aug 01 '19 at 17:28

Bill Dubuque · Answer 2 · 2021-03-26T08:06:38.720

Hint: $\,a\bmod b\,$ arises by (repeatedly) ${\rm\color{#c00}{subtracting}}\ b$ from $a,\,$ but multiples of $d$ are closed under $\rm\color{#c00}{subtraction}$.

Remark $ $ If you are familiar with groups or ideals you should find it instructive to view this from that more abstract viewpoint. Sets of (common) multiples are propotypical examples of ideals, being closed under subtraction and scaling (or, equivalently, under linear combinations).

Why can we say that if $d$ divides both $a$ and $b$, then it also divides the remainder of $a/b$?

2 Answers2