Question

What is the last digit in the number 7^1000

Answer 1

\[7^{\alpha}=****1 \iff \alpha=0 (\mod 4)\]and 1000=0(mod 4) , so the last digits is 1

Answer 2

you can see that \[7^(1000)=7*7*7*7*7.....*7\]
to get the first digit in normal multiplication you keep multiply the values until the multiplication ends, in our case after 1000 times
we notice that \[7*7=49*7=7*343=2401*7=16807\] the cycle goes as follow 7,9,3,1,7..
so observing that the number 7 repeats after 5 times of multiplication, i.e the individual complete cycle ends at 1
1000/4=250 complete cycle, thus the last digit is 1