Track / Stage 5 / 390 of 400 #990 of 1964

Problem 990

AIME late

Number theory Difficulty 5.8 Prove it

Show that any number of the form $M=\overline{x y 3}^{1}+\overline{x y 3}^{2}+\overline{x y 3}^{3}+\ldots+\overline{x y 3}^{100}$ is divisible by 10.

(Ionuț Mazalu, Brăila problem S:E14.206 GM 9/2014)

- LOCAL STAGE 28.02.2015 -

Note: Each problem is scored from 0 to 7 points.

Any other solution is assimilated according to the scoring guidelines.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

## Subject 4.

Solution Details	Associated Scoring
!	$3 \mathrm{p}$
- decompose M into the sum of 25 numbers divisible by 10 .......................	$4 \mathrm{p}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.