I do not really understand how to find the least non negative residue for an integer n mod m. Can you anyone explain to me with an example how to do it?
finding the least nonnegative residue
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congruences
congruence-relations
reduced-residue-system
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0Least nonnegative residue of $n$ mod $m$ is literally the remainder you get when you divide $n$ by $m$. For example, the least residue of $30$ mod $7$ is $2$. – 2017-02-22
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0@D_S okay so if I am asked to find the least nonnegative residue for n =3^1000 and m= 7 how would I approach that? – 2017-02-22
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0There are many ways. One approach would be to use Fermat's little theorem, which says that $3^6 \equiv 1 \mod 7$ – 2017-02-22
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0@joeb would that be my final answer? or should I reduce it more ? is another way ? like with more steps and more algebra? – 2017-02-22
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0No, that would only tell you the least non-negative residue for $3^6$, not necessarily for $3^{1000}$. You should use the property that, modulo $m$ if $a \equiv b$ then $a^k \equiv b^k$. For instance, $3^{12} = (3^6)^2 \equiv (1)^2 = 1 \mod m$. Can you see how to chew away at $3^{1000}$? – 2017-02-22
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0Hint: $1000 = 6 \cdot 166 + 4$ – 2017-02-22
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0@joeb I think these are enough to get me started.Thank you! – 2017-02-22