Lets say $a = 14, b = 20, m = 6$
$a \equiv b \pmod m $
$ 14 \equiv 20 \pmod 6$
$14 \equiv 2 $ is not true?
Because $20 \pmod 6 = 2$?
What am I doing wrong?
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it is, $14\pmod 6 = 2$ – Phicar Dec 07 '16 at 00:57
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Did you mean to have a $\pmod 6$ on the line with $14 \equiv 2$? It seems like you meant it to be there. – Ross Millikan Dec 07 '16 at 00:58
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$14 = 2\times 6 +\color{blue}{2}$, and $20 = 3 \times 6 + \color{blue}{2}$, and $2 = 0\times 6 + \color{blue}{2}$. $\quad 2, 14, 20;$ are all congruent, modulo $6$. (As is $8 = 1\times 6 + 2$...etc) – amWhy Dec 07 '16 at 01:04
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You might like to read over this question and answers – Joffan Dec 07 '16 at 01:13
2 Answers
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$14 \equiv 2 \pmod 6$ is true. $14-2=12$ is divisible by $6$. Why do you think it is not true?
Ross Millikan
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Oh i didn't realize the mod m applied to both sides? Wait so what is the final answer on both sides? 2 is equivalent to 2? – James Mitchell Dec 07 '16 at 01:00
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$\pmod m$ defines what we mean by equivalent. I would say it applies to the equivalence sign, not to either side of it. You can reduce each side to a representative, which we usually take to be one of $0,1,2,\ldots m-1$ and here is $2$. – Ross Millikan Dec 07 '16 at 01:03
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So is $a \equiv b \pmod m$ the same as $a\mod m \equiv b \mod m$? – James Mitchell Dec 07 '16 at 01:17
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I have only seen the left version in math. Computers regard mod as an operator and often write it $%$. In that case one should write the second as $a%m=b%m$ or $a \bmod m = b \bmod m$ because you are computing the representative, but it is a fine point. I think most people would accept your second version. – Ross Millikan Dec 07 '16 at 01:55
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$14$ and $20$ differ by a multiple of $6$. So do $14$ and $2$. All three are congruent modulo $6$.
Ethan Bolker
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So is $a \equiv b \pmod m$ the same as $a\mod m \equiv b \mod m$? – James Mitchell Dec 07 '16 at 01:18
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@JamesMitchell Yes, although it's not usually written that way. You'd probably say $a \pmod m = b \pmod m$ if you wrote it at all. Just $a \equiv b \pmod m$ is standard. – Ethan Bolker Dec 07 '16 at 14:01