What will be the units digit of $7777^{8888}$? What will be the units digit of $7777$ raised to the power of $8888$ ? 
Can someone do the math with explaining the fact "units digit of $7777$ raised to the power of $8888$"?

$$7777^{1} \equiv 7 \pmod{10}$$
$$7777^{2} \equiv 9 \pmod{10}$$ 
$$7777^{3} \equiv 3 \pmod{10}$$
$$7777^{4} \equiv 1 \pmod{10}$$
$$7777^{5} \equiv 7 \pmod{10}$$
And the relation continues, in general you see that:
$$7777^{4n + 1} \equiv 7 \pmod{10}$$
And
$$7777^{4n} \equiv 1 \pmod{10}$$
$$\frac{8888}{4} = 2222$$
Hence, $7777^{8888} \equiv 1 \pmod{10}$.
