## Last Digit

What would the units (ones) digit be for the value of 3 raised to the 9999th power?
Source: mathcontest.olemiss.edu 3/30/2009

SOLUTION
$3^0=1$
$3^1=3$
$3^2=9$
$3^3=27$
$3^4=81$
$3^5=243$
$3^6=729$
$3^7=2187$
$\cdots$

The pattern of the units digit is 1, 3, 9, 7 and it repeats in groups of 4. If we divide a power by 4, the remainder of the division will tell us which one of the 4 the units digit will be.

If the remainder is 0, the units digit is 1.
If the remainder is 1, the units digit is 3.
If the remainder is 2, the units digit is 9.
If the remainder is 3, the units digit is 7.

For example, to find the units digit of $3^9$, we divide the power 9 by 4 giving us a remainder of 1. Thus, the units digit of $3^9$ is 3.

Check $3^9=19683$.

To find the units digit of $3^{9999}$ we divide the power 9999 by 4 giving us a remainder of 3. Thus, the units digit of $3^{9999}$ is 7.