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Using the other than the units digit division method, how would I demonstrate $999^{2016}$ 's remainder when it is divided by 5?

I have decided that the best method in solving the problem was to engage with a binomial theorem as demonstrated below:

$$(1000-1)^{2016}$$

but then from here I didn't know how to proceed and I really would like to use the binomial theorem method

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  • $\begingroup$ Hint: $\;999 \equiv -1 \bmod 5\,$ and $\,(-1)^{2n} = 1\,$. $\endgroup$ – dxiv Jun 13 '17 at 4:31
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    $\begingroup$ Actually, this method works. All your terms are multiples of $1000$, hence of $5$, except for one term, which is $(-1)^{2016}$. However, people who know modular arithmetic will generally use that method instead. $\endgroup$ – user49640 Jun 13 '17 at 4:33
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$$(x+y)^n=\sum_{k=0}^n\binom{n}{k}x^ky^{n-k}$$

so $$(1000-1)^n=\sum_{k=0}^n\binom{n}{k}1000^k(-1)^{n-k}$$ all terms except the last is divisible by $1000$, thus $(1000-1)^n\equiv(-1)^n\mod5$

Actually, it's clear that $(1000-1)^n\equiv(-1)^n\mod1000$, so the last $3$ digits of $999^n$ are $999$ when $n$ is odd, and $001$ when n is even.

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$$(1000-1)^{2016}=\\\left(\begin{array}{c}2016\\ 0\end{array}\right)1000^{2016}+\left(\begin{array}{c}2016\\ 1\end{array}\right)1000^{2015}(-1)^{1}+\left(\begin{array}{c}2016\\ 2\end{array}\right)1000^{2014}(-1)^{2}+...+\left(\begin{array}{c}2016\\ 2015\end{array}\right)1000^{1}(-1)^{2015}+\left(\begin{array}{c}2016\\ 2016\end{array}\right)1000^{0}(-1)^{2016}=\\\to\\\underbrace{\left(\begin{array}{c}2016\\ 0\end{array}\right)1000^{2016}+\left(\begin{array}{c}2016\\ 1\end{array}\right)1000^{2015}(-1)^{1}+...+\left(\begin{array}{c}2016\\ 2015\end{array}\right)1000^{1}(-1)^{2015}}_{5k}+\left(\begin{array}{c}2016\\ 2016\end{array}\right)1000^{0}(-1)^{2016}=\\5k+\left(\begin{array}{c}2016\\ 2016\end{array}\right)1000^{0}(-1)^{2016}=\\5k+1$$

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Units digits are pretty simple.

Establish a pattern.

The units digits of the powers go $9$, $1$, $9$, $1$...

So $999^{2n}$ or any even power ends in a $1$.

If it ends in a $1$, then it is $1$ mod $5$, and the remainder is $\boxed1$

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