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I was given this question

A store sold $72$ decks of cards for $\$a67.9b$, where $a$ and $b$ are digits. Find $a + b$.

I'm not really sure how to solve this. I tried $100a + 67.9 + b \cdot0.01 \equiv 0 \pmod{72}$. From there you can get $100a + b \cdot 0.01 \equiv 4.1 \pmod{79}$. But this can't work. $b$ is maximum $9$, so you'll always end up with something in the hundreds place when you've divided. What am I supposed to do to solve?

Aiden Chow
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  • $!\begin{align} &\bmod 8!:\ 0 \equiv a679b \equiv790+b \equiv 6+b\Rightarrow b\equiv 2\Rightarrow \color{#c00}{b = 2},,\ \text{by },\ 0\le b\le 9\ &\bmod 9!:\ 0 \equiv a679b \equiv a!+!6!+!7!+!9!+!b\equiv a!+!\color{#c00}b!+4 \equiv a!+!6\Rightarrow a\equiv 3\Rightarrow a=3\end{align}\ \ $ – Bill Dubuque Dec 10 '20 at 18:25
  • We used casting out nines in the prior line, and in the first line we used that $,\color{#c00}{8\mid 1000},,$ by $,2\mid 10\Rightarrow 2^3\mid 10^3,,$ so $\bmod \color{#c00}8!: a6(\color{#c00}{1000})+79b\equiv 79b\ \ $ – Bill Dubuque Dec 10 '20 at 18:31

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