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I am trying to prove a different equation and am able to if the following is true, but I am not exactly sure if it is true.

If $a\equiv b \pmod{9}$ and $c+d = 9$, is $ca+bd \equiv 0 \pmod{9}$ a true statement?

I have tried various examples, like

$a = 29$, $b = 2$, $c = 2$, and $d = 7$

$a = 13$, $b = 4$, $c = 4$, and $d = 5$

and more.

If anyone has a counterexample, please let me know! Otherwise, if this is true, then can someone please explain why? Thanks!

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    $\begingroup$ Use $a=b+9k$ in $ca+bd$ $\endgroup$
    – Empy2
    Jul 15 '19 at 0:41
  • $\begingroup$ Hint: eliminate $\,a\,$ & $\,c\,$ and apply congruence sum & product rules - see my answer. $\endgroup$ Jul 15 '19 at 1:04
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If $a\equiv b\pmod n$, then this is just $$a(c+d) \equiv an\equiv 0\pmod n$$

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$a\equiv b\pmod{9}$ and $c+d=9$ implies $a-b\equiv 0\pmod{9}$ and $d\equiv -c \pmod{9}$ ,respectively. So

$$ca+bd\equiv ca-cb\equiv c(a-b)\equiv c\cdot 0\equiv 0 \pmod{9}.$$

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$$ \begin{align}&\ \ \ \ \ \color{#c00}a\ \equiv\,\ \color{#c00}b\\ &\ \ \ \ \ \color{#0a0}c\ \equiv \color{#0a0}{-d}_{\phantom{|}}\\ \hline {\Longrightarrow}\ \ &\ \ \ \color{#0a0}c\ \color{#c00}a\,+\,d\,b\\[.2em] {\equiv}\ \ \ & \color{#0a0}{-d}\,\color{#c00}b\,+\,d\,b\,\equiv\, 0\end{align}\qquad$$

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    $\begingroup$ We eliminated $\,a\,$ & $\,c\,$ using the Congruence Sum & Product Rules $\ \ $ $\endgroup$ Jul 15 '19 at 1:00
  • $\begingroup$ Do you manually insert the code like \color{#c00} when needed or do you have a tool for coloring conveniently? $\endgroup$
    – user9464
    Jul 25 '19 at 0:03
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    $\begingroup$ @Jack Sometimes I use tools (AHK,Emacs,etc) sometimes by hand (depends on the device). $\endgroup$ Jul 25 '19 at 1:45
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Just to list another approach: $$ ac+bd = a\underbrace{(c+d)}_{\text{multiple of $9$}} + \underbrace{(b-a)}_{\text{multiple of $9$}}d, $$ (this is an equal sign, not a congruent sign) so the left-hand side must be a multiple of $9$, too.

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