Find the last three digits

[tanmoy]

What are the last three digits of the number $7^{9999}$

[nishat protyasha]

$$7^4 \equiv 1 \pmod 8 \Rightarrow 7^{10000}\equiv1\pmod8 $$.
$$7^{100}\equiv1\pmod{125} \Rightarrow 7^{10000}\equiv1\pmod{125} $$.
$$7\cdot 7^{9999}\equiv1 \pmod{1000}\Rightarrow7^{9999}\equiv143\pmod{1000}$$

[Masum]

$$7^4 \equiv 1 \pmod 8 \Rightarrow 7^{10000}\equiv1\pmod8 $$.

Note that, also $7^2\equiv1\pmod8$ too. This helps in some cases. I mean, in such cases, try to find the smallest such integer if possible.

[mutasimmim]

I may add a little to to Masum vai. Even if you don't want to find the least integer, you can often find a "very" smaller number for composite modules using Carmichael Function.
http://en.wikipedia.org/wiki/Carmichael_function