CAT 2022 Slot 3 QA Question 4: A school has less than 5000 students and if the students are divided equally into teams of either 9 or 10 or 12 or 25 each, exactly 4 are always left out. However, if they are divided into teams of 11

Question

A school has less than 5000 students and if the students are divided equally into teams of either 9 or 10 or 12 or 25 each, exactly 4 are always left out. However, if they are divided into teams of 11

Accepted Answer

Let the number of students be $S$.
Since $S$ leaves remainder 4 when divided by 9, 10, 12 and 25, $S - 4$ must be divisible by their LCM.
$\mathrm{LCM}(9, 10, 12, 25) = 900$
$S = 900k + 4$
Also, $S$ is divisible by 11.
$900k + 4 \equiv 0 \pmod{11}$
$9k + 4 \equiv 0 \pmod{11}$
$9k \equiv 7 \pmod{11}$