3. Consider the number $N=9+99+999+\ldots+\underbrace{999}_{2014 \text { digits }}+2014$

a) Prove that the number $N$ is divisible by 10;

b) Determine the remainder of the division of the number $N$ by 111.

Supplement to Gazeta Matematică, November 2013
3. a) The last digit of $N$ is 0, and from here the conclusion;

$3 p$
b) $N=10-1+100-1+1000-1+\ldots+\underbrace{1000.0}_{\text {2015 digits }}-1+2014=\underbrace{111 \ldots 110}_{2015 \text { digits }}$.

$2 p$

The number $\underbrace{111 . .1100}$ is divisible by 111, so the remainder of the division is 10.

2p

2015 digits