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If the positive integer $x$ leaves a remainder of $2$ when divided by $8$, what will the remainder be when $x + 9$ is divided by $8$?

I love to put stuff into algebraic equations to make life easier; however, this simple (or at least that what it appears to be) problem stumps me (specifically the first part, regarding the remainder). How would I express this algebraically?

Thanks! Sorry if this is a stupid question, but I'm studying to improve my math.

user171213
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4 Answers4

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The easiest way:

Note that $x-2$ is a multiple of $8$, hence $x-2=8k,\ k\in\mathbb{Z}.$

Therefore $$x+9=(x-2)+11=(x-2)+8+3=8k+8+3=8(k+1)+3.$$

Now we can conclude the remaind is $3$.

The cool way:

We have that $$x\equiv2\ (mod\ 8)$$ Add $9$ in both sides we have $$x+9\equiv11\equiv3\ (mod \ 8).$$

DiegoMath
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Using notation more popular in Computer Science than in Mathematics, we could say $x\bmod 8=2$. What is $(x+9)\bmod 8$?

We have $9\bmod{8}=1$.

But in general $$(a+b)\bmod{m}=(a\bmod{m}+b\bmod{m})\bmod{m}.$$ It follows that $$(x+9)\bmod 8=(2+1)\bmod{8}=3.$$

Remark: An alternate computer oriented notation for $y\bmod m$ is $y\operatorname{\%}m$.

André Nicolas
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  • Thank you for the response! "mod" trick is really neat! – user171213 Aug 23 '14 at 03:03
  • You are welcome. I should stress that I was using notation that is common in Computer Science, often with $%$ instead of $mod$ (shorter!). The number-theoretic meaning of mod is related, but different. The answer by DiegoMath uses mod in the sense that is standard in Mathematics. – André Nicolas Aug 23 '14 at 03:07
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$\begin{eqnarray}{\bf Hint}\qquad\ x &\!=\!& \ \ 2 + 8q\\ \Rightarrow\,\ 9+x &\!=\!& 11+8q\, =\, 3 + 8(1\!+\!q)\ \ \text{has remainder}\,\ 3\end{eqnarray}\ $

Or, $\,\ {\rm mod}\ 8\!:\,\ \begin{array}{}\color{#c00}{9\equiv 1}\\ \color{#0a0}{x\equiv 2}\end{array}\,\Rightarrow\, \color{#c00}9+\color{#0a0}x\equiv \color{#c00}1+\color{#0a0}2\equiv 3\,$ by the Congruence Sum Rule.

Community
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Bill Dubuque
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I assure you this is definitely the most easiest answer for this question.

Q. If x leaves remainder 2 when divided by 8, what will the remainder be when x+9 is divided by 8?

x/8 gives remainder of 2

eg:10/8 gives remainder of 2


to get remainder of x+9/8 :


substituting x=10  in x+9/8:


19/8 gives remainder 3,
so the answer is 3.


its as simple as that!:)
smeaky
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    Please do not answer 8 year old questions that already have perfectly good accepted answers unless you have something truly insightful and unique to say. Also, in the future, please format your answers using MathJax. – Christian E. Ramirez Oct 23 '22 at 02:37