The Power of Powers

[Debasis]

228^251 * 126^23416 * 37^217 * 213^19. what will be the last digit of the resulting Answer?

[mungbeans]

Spoiler for

6

[BobbyGo]

228^251 * 126^23416 * 37^217 * 213^19. what will be the last digit of the resulting Answer?

The short answer: 8

The long answer: The last digit of any number raised to a power will be equal to the last digit of a similarly ending number raised to the same power. (the last digit of 4^5 will equal the last digit of 14^5 will equal the last digit of 5374^5 and so on) Since we are only interested in the last digit of the resulting answer, the equation can be condensed to:

8^251 * 6^23416 * 7^217 * 3^19

Additionally, the same concept can be applied to multiplying different numbers together (in the context of finding the last digit). For example, the last digit of 7 * 3 will equal the last digit of 67 * 123 will equal the last digit of 887 * 76543 and so on, so only the last digit of each segment of the equation is needed to be multiplied.

Last digit of 8^251 = 2

Last digit of 6^23416 = 6

Last digit of 7^217 = 7

Last digit of 3^19 = 7

Last digit of 2 * 6 * 7 * 7 = 8

Edited March 14, 2013 by BobbyGo

[bonanova]

The last digit of a number raised to a power cycles with a given periodicity.

8=> 8 4 2 6 8 4 ... every 4 powers it repeats
6 => 6 6 6 6 6 ... always ends in 6
7 => 7 9 3 1 7 9 ... repeats every 4 powers
3 => 3 9 7 1 3 9 ... repeats every 4 powers.


251 mod 4 = 3 so first term ends in 2
23416 mod 1 = 1 so second term ends in 6
217 mod 4 = 1 so third term ends in 7
19 mod 4 = 3 so fourth term ends in 7

Product of 2 6 7 7 is 588

so last digit of the expression is 8.