Question

What is the final digit in the number 2687^398? Hint: First compute 7^1, 7^2, 7^3, 7^4, 7^5,..., until you see a pattern. Why are the final digits of these numbers the same as the final digits of 2687^1, 2687^2, 2687^3....,?

Answer 1

9 from the actual expansion of 2687^398 by Mathematica, a program capable of unlimted precision arithmetic.
The expansion is attached.

Answer 2

Okay, in class though, one person said that you could do it based on a pattern with the different powers of 7, do you know anything about that?

Answer 3

@robtobey

Answer 4

use the given hint

Answer 5

consider last digits of powers of 7 :

7^1 : 7
7^2 : 9
7^3 : 3
7^4 : 1

7^5 : 7
7^6 : 9
7^7 : 3
...

see any pattern ?