CAT 2022 Slot 3 QA Q19 Number Systems

Question 19 of the Quantitative Ability (QA) section, CAT 2022 Slot 3, with CAT's official answer and a worked solution.

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 each, no one is left out. The maximum number of teams of 12 each that can be formed out of the students in the school is

TITA: type in the answer. There are no options and no negative marking.

Answer: 150 Number Systems TITA

Solution

  1. \(N-4\) is divisible by \(\text{lcm}(9,10,12,25)=900\), so \(N=900k+4\lt5000\).
  2. Need \(11\mid N\): \(900\equiv9\pmod{11}\), so \(9k+4\equiv0\Rightarrow k\equiv2\pmod{11}\). Only \(k=2\): \(N=1804\).
  3. Teams of 12: \(\lfloor1804/12\rfloor=150\).

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