If a number n is divisible by 3, 5, and 12, then calculate the next largest…
2024
If a number n is divisible by 3, 5, and 12, then calculate the next largest number that is also divisible by 3, 5, and 12.
- A.
n + 60
- B.
n + 34
- C.
n + 23
- D.
n + 12
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Show answer & explanation
Correct answer: A
Concept: A number is divisible by several given integers exactly when it is divisible by their Least Common Multiple (LCM). Multiples of the LCM repeat at a fixed interval equal to the LCM itself, so the next number after n divisible by all of the same integers is n + LCM.
Application: The three divisors here are 3, 5, and 12.
Factorise 12 = 22 × 3, so the LCM of 3, 5, and 12 needs the factors 22 (from 12), 3 (common to 3 and 12), and 5.
So LCM(3, 5, 12) = 22 × 3 × 5 = 60.
Since n is divisible by 3, 5, and 12, n must be a multiple of 60 — write n = 60k for some integer k.
The next number divisible by all three is n + 60 = 60k + 60 = 60(k + 1), which is again a multiple of 60.
Cross-check: None of the other increments (34, 23, 12) is a multiple of 60 — for instance, 12 alone is not divisible by 5 — so adding any of them would not preserve divisibility by 3, 5, and 12. Only n + 60 keeps the result divisible by all three.