If n is a product of two unequal prime numbers, then it has exactly 4 divisors.

Step-by-step solution

Idea: A divisor of pq can use each of the primes p and q at most once, giving 1, p, q, pq. But a cube of a prime, p³, also has 4 divisors: 1, p, p², p³.

1. Converse: If n has exactly 4 divisors, then it is a product of two unequal prime numbers.½ mark
2. Proposition — true. Let n = pq with p, q different primes. A divisor of pq can contain p at most once and q at most once, so the divisors are 1, p, q, pq — four different numbers. E.g. 15: 1, 3, 5, 15.1 mark
3. Converse — false. Counterexample: n = 8. Its divisors are 1, 2, 4, 8 — exactly 4.½ mark
4. But 8 = 2 × 2 × 2 is not a product of two unequal primes (its only prime factor is 2). So the converse is false.½ mark
5. Proposition true; converse false. (Every cube of a prime, such as 27 = 3³, is also a counterexample.)½ mark

Answer to write in the exam

Common mistakes that cost marks

• Testing the converse with 6, 10 or 15. They are products of two unequal primes, so they agree with the converse.
• Forgetting that 1 and n itself are divisors when counting.

How this can come in the exam

Try one yourself

How many divisors does 4 have? Does 4 break the proposition?

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