Question 913080: I need help with my homework. How do I solve this:
What is the units digit of 13^162
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! 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 9 (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.
Feel free to email me at jim_thompson5910@hotmail.com
or you can visit my website here: http://www.freewebs.com/jimthompson5910/home.html
Thanks,
Jim
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