I have an assignment where the following statement is:
Determine the lowest positive integer that is congruent with the statement mod 12: $$ 8^{2012} + 2012^8 $$
How can I solve this? I have totally forgotten...
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3Compute mod $4$ (instant) and mod $3$. – André Nicolas Nov 25 '15 at 18:52
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why does that work? – rablentain Nov 25 '15 at 18:53
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1There are many possible strategies. But in this case, factoring $12$ is useful, since we can do calculations mod $4$ and mod $3$ with minimal calculation. – André Nicolas Nov 25 '15 at 18:57
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1we get $$8^{2012}+2012^{8}\equiv 8 \mod 12$$ – Dr. Sonnhard Graubner Nov 25 '15 at 19:13
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@theva: You can use that $4^n\equiv 4\pmod{12}$ for every positive integer $n$ if you can prove that. – mathlove Nov 25 '15 at 19:21
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prove it by induction for example. – DanielWainfleet Nov 25 '15 at 19:45
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We have $\;8^n\equiv\begin{cases}4\mod12&\text{if $n$ is even},\\8\mod12&\text{if $n$ is odd}\end{cases}.$
Furthermore $\;2012\equiv 8\mod 12$, hence $$8^{2012}+2012^8\equiv 4+4=8\mod 12.$$
Bernard
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