Sufficiently Many Eights

Number Theory Level 4

$\Large \underbrace{88888888888\ldots 8}_{888\text{ number of 8's}}$

Find the remainder when the above number is divided by 887.

1 solution

Akshat Sharda
Mar 12, 2016

$\underbrace{888888\ldots 88}_{888\text{ times}}=8\cdot \frac{(10^{888}-1)}{9}$

We know, $10^{\phi(887)}=10^{886}\equiv 1\pmod{887} \\ \therefore 8\cdot \frac{(10^2-1)}{9}=8\cdot 11=\boxed{88}$

$• \ 887 \text{ is a prime number.}$

clear solution

Soner Karaca - 5 years, 2 months ago

Great solution.

Mehul Arora - 5 years, 2 months ago