There are 72 students and 15 teachers in a school excluding principal. The…
2024
There are 72 students and 15 teachers in a school excluding principal. The principal has 3-digits odd no. quantities of .toffees and he equally distributes it among students & teachers, he gets 86 Toffees remain. If he distributes among teachers, he gets 14 Toffees remain. If principal adds himself among teachers, how much coffee(s) will remain?
- A.
1
- B.
5
- C.
15
- D.
Can not be determined
Attempted by 25 students.
Show answer & explanation
Correct answer: B
Step 1: Translate the conditions into congruences.
There are 72 students and 15 teachers, so distributing among students and teachers together means dividing by 72 + 15 = 87. The remainder is 86, so
N ≡ 86 (mod 87).
Distributing among teachers (15) leaves remainder 14, so
N ≡ 14 (mod 15).
Step 2: Solve the two congruences simultaneously.
Note gcd(87,15) = 3 and both remainders are congruent modulo 3, so solutions exist.
Combine the congruences to get a single modulus: lcm(87,15) = 435, and the combined congruence is
N ≡ -1 (mod 435),
So general solution: N = 435t - 1 for integer t.
Step 3: Find a 3-digit odd value of N.
Try t = 2: N = 435·2 - 1 = 869, which is a 3-digit odd number and satisfies both original conditions.
Step 4: Include the principal among teachers and compute the remainder.
Teachers plus principal = 15 + 1 = 16. Divide 869 by 16: 869 = 16×54 + 5, so the remainder is 5.
Answer: 5