Simple Math Problem

[Jim Brier]

If you take 513 and raise it to the 128th power, what is the one's digit? Is there a simple way of figuring it out?

If you only care about the one's digit, it only matters that 513 ends with a 3.

3^0 ends with 1
3^1 ends with 3
3^2 ends with 9
3^3 ends with 7

from here, it repeats: 1, 3, 9, 7. So simply take the power and divide by 4. The remainder should tell you which of these is the last digit.

For your example: 128 mod 4 is 0, therefore 513^128 ends with a 1.

Hope that helps!

[M.B.E.]

"Anonymous" is correct. Notice that 3^4=81 so 3^128=81^32. Now when you take 81 to any whole power the ones digit can only be 1, since 1x1=1. The ones digit of 513^128 is the same as the ones digit of 3^128, which is 1.

[Jim Brier]

(n/t)

last semester i was in a number theory class at university, and i thought to myself while these very questions were being brought up in class...."who the hell would want to know that" well now i know

thanks for making my class not an entire waste

[irchans]

For many years number theory had no applications but since the advent of computers, many applications have arrived. Most of the applications are in encryption, but you occasionally run into number theory in other fields (e.g. CDMA for cell phone transmission, pseudo-random number generation).


PS:
513^128 =

78527674095678372984552781105929458007589456144486 82
65644231223842532332696067633795458244587461783802 15
02477421627930646460271928516001276777242677537671 02
04753715518451952675490654125302312881638160106707 57
06828700240755793013636138691904689489491326625021 94
48497782634590705850754151909775421521223364755619 37
93612817448994130470020200985067521

[mdlm]

What software did you use to do this computation?

[irchans]

Mathematica

www.wolfram.com (http://www.wolfram.com)

(Student discounts are substantial).