# Ask Uncle Colin: A Thirteenth Root

Dear Uncle Colin,

What’s the 13th root of 21,982,145,917,308,330,487,013,369? I know it’s an integer.

- Extremely Large Exponent, Perhaps Having A Nice Thirteenth…

Hi, ELEPHANT, and thanks for your message!

Thirteenth roots are a staple of mental arithmetic contests (I’m told ), but despite this being apparently a big number, it’s not *that* big.

It’s a 26-digit number, so it’s smaller than $10^{26}$, which is $100^{13}$ – so the root we’re after is smaller than 100.

### The last digit

Any odd number raised to the fourth power ends in a 1. As a result, any odd number raised to the twelfth power also ends in a 1, and any odd number raised to the 13th power ends in the same digit as the result. In our case, the number we want ends in a 9.

*whoosh*

Hello, sensei, I’ve been expecting you.

“Curse your fiendish trap! The base ten log of this number is about 25.3, or (26 - 0.7) and a thirteenth of that is 2 - 0.05. Also, $\log(9)$ is 0.95 or so; the root is about 90 and exactly 89.”

Thank you, sensei.

### But we don’t need logs

Thinking modulo 100, and knowing the root ends in a 9, $(10a - 1)^{13}$ is congruent to $(130a - 1)\pmod {100}$, or even $(30a -1)$. I need to solve $30a \equiv 70 \pmod{100}$, and $a=9$ is an obvious answer (and the only one for $a < 10$).

So, $a = 9$ and the root is $90 - 1 = 89$.

Hope that helps!

- Uncle Colin