Track / Stage 2 / 185 of 240 #425 of 2444

Problem 425

Number theory Difficulty 2.7 Multiple choice CEMC Gauss (Grade 7) · Canada · 2026

Emil chooses a positive integer so that

It is between $100$ and $199$ inclusive.
It is divisible by $4$ .
The sum of its digits is $5$
or more.

How many different such integers could Emil choose?

Pick one

Official solution

Between $100$ and $199$ inclusive, the numbers that are
divisible by $4$ are: $4\times25=100, 4\times26=104, 4\times27=108, 4\times28=112, \dots, 4\times49=196$ Thus, there are $49-25+1=25$ such numbers.

Of these, $100$ , $112$ and $120$ are the only numbers whose digit sum
is less than $5$ .

Therefore, between $100$ and $199$ inclusive, there are $25-3=22$ numbers that are divisible by
$4$ and for which the sum the digits
of each number is $5$ or more.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.