Breaking Down an Exponent Question

by , Aug 18, 2009

Periodically, I'm going to pull up a particular problem that was discussed on the forums and share the discussion and solution here. I ran across this (old!) post recently but the shortcut here is still very much relevant.

First, here's a link to the original thread: https://www.beatthegmat.com/exponent-problem-t1264.html

Here's the problem:

What is the remainder when 43^43+ 33^33 is divided by 10?

No solutions were provided with the problem. You've already tried the problem, now, right? You're not trying to read the solution before you've actually tried it? If not, go try it right now - don't read any further yet. :)

Okay, so now that you've tried it, let's discuss.

Whenever they ask you a "units digit" question with really high exponents, there will be a pattern and you will only have to follow the units digit through the problem (because it is all multiplication and, in multiplication, the units digits of the two numbers determine the value of the units digit of the product). So your task now is to find that pattern.

Our first number is 43^43. The units digit of 43^1 = 3. The units digit of 43^2 = 3*3 = 9. The units digit of 43^3 = 3*3*3 = 27 = 7. The units digit of 43^4 = 3^4 = 81 = 1. The units digit of 43^5 = 3^5 = 243 = 3.

Bingo! Here's the pattern: 3, 9, 7, 1, 3, 9, 7, 1, ...

If you're feeling ambitious, you can memorize this pattern. Any number ending in a units digit of 3 will have the units digit pattern: 3, 9, 7, 1, repeating.

The pattern repeats every 4th term. So 3^4, 3^8, 3^12, etc, will all have the units digit 1. What is the largest multiple of 4 that is still less than the exponent, 43, in our starting number? 40 is the largest multiple of 4 that is still smaller than 43. So, 3^40 will have a units digit of 1, 3^41 will have a units digit of 3, 3^42 will have a units digit of 9, and 3^43 will have a units digit of 7.

Do the same thing for our second number, 33^33. Go ahead, do it now before you keep reading. :)

Same pattern as above. 3^32 will have a units digit of 1, 3^33 will have a units digit of 3.

So the units digit of 43^43 = 7 and the units digit of 33^33 = 3. 7 + 3 = 10, which is a units digit of 0. Anything that ends in 0 will also have a remainder of 0 when divided by 10!