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 $8888$"?

the units digit of a number is the same as the number mod 10
so we just need to compute $7777^{8888} \pmod {10}$
first $$7777^{8888} \equiv 7^{8888} \pmod {10}$$ and secondly by Eulers totient theorem $$7^{8888} \equiv 7^{0} \equiv 1 \pmod {10}$$ by Eulers totient theorem (since $\varphi(10)=4$ and $4 \mid 8888$)
so there's a quick way to see that the last digit is a 1
