#1**0 **

8^25 = 377 7893186295 7161709568 +

12^25 = 9539621 6644069012 9601298432

**=9539999 4537255308 6763008000**

Guest Oct 20, 2018

#2**+1 **

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.

tertre
Oct 20, 2018

#7**0 **

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**0 **

But 12^5 DOES NOT END IN ZERO =248,832 !!

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

Guest Oct 20, 2018

#4**0 **

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

SaltyGrandma
Oct 20, 2018

#9**0 **

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**+1 **

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

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

Melody
Oct 21, 2018