In a large school auditorium, the students are made to sit to watch the…
2024
In a large school auditorium, the students are made to sit to watch the program. If the teachers make a row of students of 16 each, there will be 12 students left. If they make rows of 24 each, then there will be 20 students left, if they make rows of 25 each, there will be 21 students left and if they make rows of 30 each, there will be 26 students left. What is the minimum number of students present in the school?
- A.
1216
- B.
1784
- C.
1196
- D.
2396
Attempted by 857 students.
Show answer & explanation
Correct answer: C
Let the total number of students be N. The conditions mean:
N = 16k + 12, so N + 4 is divisible by 16.
N = 24m + 20, so N + 4 is divisible by 24.
N = 25p + 21, so N + 4 is divisible by 25.
N = 30q + 26, so N + 4 is divisible by 30.
So N + 4 must be a common multiple of 16, 24, 25, and 30. The least such multiple is their LCM.
Compute the LCM:
16 = 2^4, 24 = 2^3 × 3, 25 = 5^2, 30 = 2 × 3 × 5
LCM = 2^4 × 3 × 5^2 = 1200
Minimum N + 4 = 1200, hence minimum N = 1200 − 4 = 1196.