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# What is the sum of the last two digits of​ 8^{25}+12^{25}?

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What is the sum of the last two digits of $$8^{25}+12^{25}$$?

Oct 20, 2018

#1
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8^25 = 377 7893186295 7161709568  +

12^25 = 9539621 6644069012 9601298432

=9539999 4537255308 6763008000

Oct 20, 2018
#2
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Hey, SG here is a short way!

We know that the square root of 25 is 5, so every 5th power should end in the last digit.

Thus, 12^5 ends in a 0, and 8^5 also end in a zero. Now, we have two zeroes and doing some quick computation, we have 0+0=0.

Oct 20, 2018
#7
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But we want the SUM of the last two digits of the SUM of 8^25 + 12^25  NOT the sum of the last (single) digits of each term   .....I realize the answer may be the same, but this method is not really a true method......

ElectricPavlov  Oct 21, 2018
#3
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But 12^5 DOES NOT END IN ZERO =248,832 !!

Also, 8^5 ............................................  =32,768 !!

Oct 20, 2018
#5
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Sorry....the question asks for    8^(25)   and  12^(25)     NOT ^(5)     ~~ Guest Oct 20, 2018
#4
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0 is the answer everyone! Thanks all! The way to do it is that we find out that the expression is divisible by 100.

Oct 20, 2018
#9
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Hi Salty,

"0 is the answer everyone! Thanks all! The way to do it is that we find out that the expression is divisible by 100."

Without finding the full number first (in which case the answer would be self evident) can you please show us how you can factor out 100 from this expression.

I have verified Tertre's answer.

Tertre's logic does seem flawed, or maybe he just did not explain it very well,  but at least he attempted to explain himself.

But Salty, you have just made a statement, you have not shown anything.

Melody  Oct 21, 2018
#8
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$$8^{25}+12^{25}\\ =(8^5)^5+(12^5)^5\\ \text{now I am only interested in the last 2 digits each time}\\ 68^5+32^5\\ 68+32\\ 100$$

The last 2 digits will be 00 when I add them together I get 0

Oct 21, 2018
#10
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Melody: I think he/she means that:[12^25 + 8^25] mod 100 =0. But then, that is meaningless, since I tried the first 1,000 moduli from 1 to 1000, and got at least 40 numbers that give zero as the answer!! Of course, 1,000 is a tiny number compared to [12^25 + 8^25].

Oct 21, 2018
#11
0

Yes, it is not helpful but it is excellent that the question, asnd answers, are making you think. :)

I hope you understand my logic (from answer#8) guest#10.  If you don't you can always ask :)

Melody  Oct 21, 2018