If $(m, 10) = 1$, choose $b$ so that $10 b \equiv 1 \pmod m$. Then $n \equiv 0 \pmod m$ if and only if $n' + ba_0 \equiv 0 \pmod m$, where $a_0$ is the unit's digit of $n$, and $n'=(n-a_0)/10$. First generalize this and tell me how to extend this theorem to general divisibility tests of other numbers by a single formula or method or procedure.
Divisibility tests for all numbers
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elementary-number-theory
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0What are n' and $a_0$? – 2011-10-04
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0I suspect that $a_0$ is the last (one’s) digit of $n$ and that $n'=(n-a_0)/10$, the number that you get when you erase the last digit of $n$; is that correct? – 2011-10-04
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0@Brain M. Scott! Your suspected one is very right. The last digit is $a_0$. – 2011-10-04
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1Edited to include Brian's interpretations. – 2011-10-04