Show that the last four digits of $2013^k$ are 0001

Question

Show that a natural $k \ge 1$ exists s.t the last four digits of $2013^k$ (written as a decimal) are 0001.

I understand that k must be of the form k=4m. The last digit of 2013 is 3 and only when powered by multiply of 4 the result ends with 1.

Unfortunately, I don't know how to advance...

Please help, thanks!

Answer 1

If $2013^k$ has last 4 digits as $1$, then $$2013^k \equiv 1 \pmod {10000}$$

Since $2013$ and $10000$ are coprime, Euler's Theorem asserts that the above relation will hold true for $k = \phi(10000) = 10000\cdot\left(1-\frac{1}{2}\right)\cdot\left(1-\frac{1}{5}\right) = 4000$. In other words,

$$2013^{4000} \equiv 1 \pmod{10000}$$

In fact, Carmichael's Theorem provides a tighter result. It asserts that $k = \lambda(10000) = 500$ will work as well, i.e.

$$2013^{500} \equiv 1 \pmod{10000}$$