How many integers between 4000 and 6000 are divisible by 32, 40, 48, and 60?
Aptitude
Problems on H.C.F and L.C.M
Difficulty: Medium
Choose an option
-
A2
-
B3
-
C4
-
D5
-
E6
Answer
Correct Answer: 4
Explanation
Introduction: An integer divisible by multiple numbers must be a multiple of their least common multiple. We count how many multiples of that LCM lie in the specified interval.
Given Data / Assumptions:
- Range: 4000 to 6000 inclusive
- Divisors: 32, 40, 48, 60
Concept / Approach: Compute LCM(32, 40, 48, 60). Then count multiples in the range using ceiling and floor division.
Step-by-Step Solution:
LCM = 480 Smallest multiple ≥ 4000: 9 * 480 = 4320 Largest multiple ≤ 6000: 12 * 480 = 5760 Values: 4320, 4800, 5280, 5760 Count = 4Verification / Alternative check: Each listed value is divisible by all four divisors by construction through the LCM.
Why Other Options Are Wrong: 2, 3, and 5 do not match the exact count of LCM multiples in the interval.
Common Pitfalls: Taking pairwise divisibility instead of using the LCM, which can overcount or undercount.
Final Answer: 4