Question 1206604
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Edit: It appears that another tutor has changed the question, which seems strange. For me the initial question simply had 13^35 and nothing else. I'll keep my solution for anyone who may need it. 


I'm not sure what you mean by "sum" when there's only one item mentioned here. It appears you want to find the units digit for 13^35


When we divide by 10 and look at the remainder, that's the same as focusing only on the units digit.
Example: 108/10 = 10 remainder 8
This is where the modulo operator comes in handy.


Let's look at powers of 13 mod 10.
13^1 = 13 = 3 (mod 10)
13^2 = 169 = 9 (mod 10)
13^3 = 13*13^2 = 3*9 = 27 = 7 (mod 10)
13^4 = 13*13^3 = 3*7 = 21 = 1 (mod 10)
13^5 = 13*13^4 = 3*1 = 3 (mod 10)


In short,
13^1 = 3 (mod 10)
13^2 = 9 (mod 10)
13^3 = 7 (mod 10)
13^4 = 1 (mod 10)
13^5 = 3 (mod 10)


The pattern is 3,9,7,1 which repeats forever.
This repeating block consists of 4 items. It will mean we divide 35 by 4 to look at the remainder.
35/4 = 8 remainder 3


The "remainder 3" leads to exponent 3, so 13^35 and 13^3 have the same remainder when dividing by 10.
13^35 = 13^3 = 7 (mod 10)


Therefore, 13^35 is some very large number that ends with a 7.


Verification using WolframAlpha
<a href="https://www.wolframalpha.com/input?i=13%5E35+mod+10">https://www.wolframalpha.com/input?i=13%5E35+mod+10</a>
and
<a href="https://www.wolframalpha.com/input/?i=13%5E35">https://www.wolframalpha.com/input/?i=13%5E35</a>
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