A number is divisible by 24 if: Statements: I) The number is divisible by 3…
2025
A number is divisible by 24 if:
Statements:
I) The number is divisible by 3
II) The number is divisible by 8
Answer: C. Statements I and II together are sufficient, but neither alone is sufficient. — For any two divisors that are coprime (share no common factor other than 1), a number is divisible by their product if and only if it is divisible by each of…
- A.
Statement I alone is sufficient in answering the problem question
- B.
Statement II alone is sufficient in answering the problem question
- C.
Statements I and II together are sufficient, but neither alone is sufficient.
- D.
Both the statements even put together are not sufficient in answering the problem question
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Correct answer: C
For any two divisors that are coprime (share no common factor other than 1), a number is divisible by their product if and only if it is divisible by each of them individually. Since 24 = 3 × 8 and 3 and 8 share no common factor besides 1, they are coprime — so a number is divisible by 24 exactly when it is divisible by both 3 and 8.
Statement I alone: divisibility by 3 does not guarantee divisibility by 8, so it does not guarantee divisibility by 24 — for example, 6 and 9 are divisible by 3 but not by 24.
Statement II alone: divisibility by 8 does not guarantee divisibility by 3, so it does not guarantee divisibility by 24 — for example, 8 and 16 are divisible by 8 but not by 24.
Statements I and II together: the number is divisible by both 3 and 8. Because 3 and 8 are coprime, this means the number must be divisible by their product (equivalently their LCM), which is 24.
Independent check: 72 is divisible by 3 (72 ÷ 3 = 24) and by 8 (72 ÷ 8 = 9), and indeed 72 ÷ 24 = 3, confirming the combined condition works; while 6 (divisible by 3 only) and 16 (divisible by 8 only) each fail to be divisible by 24, confirming neither statement alone is sufficient.
So each statement alone leaves counterexamples, but together they force divisibility by 24 — the two statements are sufficient only when combined, and neither is sufficient alone.