Number System Question 4 - CAT 2022
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:
Step-by-Step Explanation
- Find the general pattern for the student count:
When the students are split into groups of 9, 10, 12, or 25, there are always 4 students left over. This means the total number of students must be a multiple of the LCM (Lowest Common Multiple) of these numbers, plus 4.
Let's find the LCM of 9, 10, 12, and 25, which is 900.
So, the total number of students follows this pattern: 900 Ă— k + 4.
- Use the remaining clues to find the exact number:
We know the school has fewer than 5000 students, and the total must divide by 11 perfectly. Let's test different integers for k:
- If k = 1: Total = 900(1) + 4 = 904. (904 does not divide by 11).
- If k = 2: Total = 900(2) + 4 = 1804. Let's check: 1804/11 = 164 exactly!
So, the school has exactly 1804 students.
- Calculate the final number of teams:
The question asks for the maximum number of teams of 12 we can make out of these 1804 students. Let's divide:
1804/12 = 150 with a remainder of 4
This means we can form a maximum of 150 full teams.
Correct Answer: 150
đź’ˇ FundaMakers Tip:
Remainder questions with multiple divisors are always LCM problems in disguise. Once you get the baseline structure (900k + 4), don't waste time with formal equations. Just plug in 1, 2, 3… for k. You will hit the correct value on the second try!