In a class there are fewer than 500 students. When the total is divided by 3,…
2024
In a class there are fewer than 500 students. When the total is divided by 3, it gives a whole number. Similarly, when it is divided by 4, 5, or 7, it also gives a whole number. Find the number of students in the class.
- A.
420
- B.
455
- C.
487
- D.
520
Attempted by 76 students.
Show answer & explanation
Correct answer: A
Key idea: any number divisible by 3, 4, 5 and 7 must be a multiple of their least common multiple (LCM).
Step 1: Find the LCM of 3, 4, 5 and 7.
Prime factors: 3 is 3; 4 = 2^2; 5 is 5; 7 is 7.
Take the highest powers of all primes: 2^2 × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420.
Step 2: Check multiples of 420 below 500.
First multiple: 420 (valid and < 500).
Next multiple: 840 (greater than 500), so not allowed.
Conclusion: The only number less than 500 divisible by 3, 4, 5 and 7 is 420.
Answer: 420