Question

What is the units digit of 2 to the 50th power

Answer 1

Notice that


`2^1=2`

`2^2=4`

`2^3=8`

`2^4=16`

`2^5=32`

so that for each power of 2, there is a pattern of period 4. This means that for integers
`k\ge0`, each of
`2^(4k)`,
`2^(4k+1)`,
`2^(4k+2)`, and
`2^(4k+3)` have the same units digit.

We can write
`50=4(12)+2`, and since
`3=4(0)+2`, it follows that
`2^(50)` and
`2^3` share the same units digits. So the units digit of
`2^(54)` is 8.