If a and b are perfect squares, then ab is a perfect square.

Step-by-step solution

Idea: Write the squares in general form to prove the proposition. For the converse, two non-squares can still multiply to a square, e.g. 2 × 8.

1. Converse: If ab is a perfect square, then a and b are perfect squares.½ mark
2. Proposition — true. Let a = m² and b = k² for whole numbers m, k. Then ab = m²k² = (mk)², a perfect square. E.g. 4 × 9 = 36 = 6².1 mark
3. Converse — false. Take a = 2 and b = 8. Then ab = 16 = 4², a perfect square.½ mark
4. But 2 and 8 are not perfect squares (1² = 1, 2² = 4, 3² = 9). So a = 2, b = 8 is a counterexample.½ mark
5. So the proposition is true and its converse is false.½ mark

Answer to write in the exam

Common mistakes that cost marks

• Testing the converse with a = 4, b = 9. Both are squares, so this agrees with the converse and proves nothing.
• Proving the proposition with one example (4 × 9 = 36). Use m²k² = (mk)² to cover every case.

How this can come in the exam

Try one yourself

Give a counterexample to ‘If a + b is a perfect square, then a and b are perfect squares’.

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