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Find the last two digits of  \(6^{100}\)

 Jun 22, 2020
 #1
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You can start by doing 6x6, which is 36. If you multiply it by 6 again, the units digit is still six. So, the last digit is always going to be 6.

 

6^1 = 6

--------------- the stuff in between the two dash lines is one cycle of the pattern

6^2 = 36

6^3 = 216

6^4 = 1296 (by the way, for these you don't have to multiply the whole thing out, you just need to multiply the last two digits of each result by 6 until you see a pattern lol)

6^5 = 7776

6^6 = 46656

-----------------

6^7 = 279936

 

And yay! We've arrived back at the start of the pattern, with the tens digit being 3. There's 99 numbers from 2-100, and 99/5 is 19R4 and the 4th tens digit in the pattern cycle is 7.

 

So the last two digits are and 6.

 Jun 22, 2020

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