Question 913080
List out the powers of 13


13^0 = 1
13^1 = 13
13^2 = 169
13^3 = 2,197
13^4 = 28,561
13^5 = 371,293


Notice the units digits (from top to bottom) are: 1, 3, 9, 7, 1, 3, ...


So we repeat the units digit once we get to 13^4. Effectively, this tells us that we repeat every 4 increases of the exponent in 13^x (x is an integer).


We can take advantage of this by dividing the exponent by 4 and looking at the remainder. For instance, 5/4 = 1 remainder 1. That remainder 1 matches up with the 1 in the exponent of 13^1


In other words, 13^5 and 13^1 have the same units digit (due to that remainder 1)


Now do 162/4 = 40 remainder 2


This tells us 13^(162) and 13^2 have the same units digit.


Therefore, the final answer is <font size=4 color="red">9</font> (since 13^2 = 169 has a units digit of 9).


Let me know if you need more help or if you need me to explain a step in more detail.
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Thanks,


Jim