If n is divisible by 24, then it is divisible by both 4 and 6.

Step-by-step solution

Idea: 24 is a multiple of both 4 and 6, so the proposition holds. But 4 and 6 share the factor 2, so the smallest number divisible by both is 12, not 24.

1. Converse: If n is divisible by both 4 and 6, then it is divisible by 24.½ mark
2. Proposition — true. If n = 24k, then n = 4 × 6k (divisible by 4) and n = 6 × 4k (divisible by 6).1 mark
3. Converse — false. Counterexample: n = 12. 12 = 4 × 3 and 12 = 6 × 2, so 12 is divisible by 4 and by 6. But 12 < 24, so 12 is not divisible by 24.1 mark
4. The reason: 4 and 6 have a common factor 2, so ‘divisible by 4 and 6’ only means ‘divisible by 12’ (their LCM), not by 4 × 6 = 24. Proposition true; converse false.½ mark

Answer to write in the exam

Common mistakes that cost marks

• Reasoning ‘divisible by 4 and 6 means divisible by 4 × 6 = 24’. That works only when the two numbers have no common factor other than 1.
• Giving 48 or 72 as a counterexample: they are divisible by 24, so they agree with the converse.

How this can come in the exam

Try one yourself

Give a counterexample to ‘If n is divisible by 10 and by 15, then n is divisible by 150′.

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