To the fifth!

Algebra Level 2

Sally is thinking of a positive integer ending with the digit A.

Joe raises that number to the fifth power.

Arnold raises Joe's number to the fifth power.

Mickey raises Arnold's number to the fifth power.

What is the last digit of Mickey's number?

A 10-A A+3 Not enough information

1 solution

Geoff Pilling
Jan 16, 2018

It turns out that if a positive integer ends with the digit A, then that number raised to the fifth power also ends with A.

Therefore, however many times you raise a number to the fifth power, the last digit will always be $\boxed{A}$

More details: Essentially, we want to show that $A^5 \equiv A \pmod {10}$ for all integers $A$ .

This is equivalent to proving the 2 statements: $A(A^4 - 1) \equiv 0 \pmod 2$ and $A(A^4 - 1) \equiv 0 \pmod 5$ .

Apply parities of integers for the first one and Fermat's little theorem for the second one.

Pi Han Goh - 3 years, 4 months ago

Ah, good point!

Geoff Pilling - 3 years, 4 months ago