This only proves Q. It doesn’t prove that every square number has an odd number of factors (Proposition P).
Is P true?

Step-by-step solution

Idea: Start from n = f × f and count factors in factor–partner pairs. The key extra fact is that a number cannot have two different repeated pairs.

1. Let n be a perfect square, n = f × f. Then (f, f) is a factor–partner pair in which the same number repeats.½ mark
2. There cannot be a second such pair (g, g). It would need g × g = f × f. If g were bigger than f, g × g would be bigger than f × f; if smaller, it would be smaller. So g = f.1 mark
3. Every other factor d has a partner n ÷ d different from d. These factors come in pairs of two different numbers, so there is an even number of them.½ mark
4. Total number of factors = (even number) + 1 (for f) = an odd number. So P is true.½ mark
5. Together with Q, this shows that ‘n is a perfect square’ and ‘n has an odd number of factors’ imply each other.½ mark

Check: 49 = 7 × 7: factors 1, 7, 49 (three). 16 = 4 × 4: factors 1, 2, 4, 8, 16 (five). Both odd.

Answer to write in the exam

Common mistakes that cost marks

• Saying ‘Q is true, so P is true’. A true converse tells us nothing about P; P needs its own proof.
• Leaving out the step that there is only one repeated pair. Without it, the count could be even.
• Checking a few squares (4, 9, 16) and stopping. Examples support the claim but do not prove it.

How this can come in the exam

Try one yourself

List the factors of 81 in factor–partner pairs. Which pair repeats a number, and how many factors are there?

More questions like this

• (i) n has an odd number of factors
  which implies
  (ii) the existence of a factor-partner pair in which the same number repeats, say (f, f)
  which implies
  (iii) n is a square number: n = f × f
  We start with (iii). Clearly, (iii) implies (ii). Does (ii) imply (i)?
• For example, if the number of such pairs is two — say (f, f) and (g, g) — what can we say about the number of factors?
• Proposition P: If two triangles have the same area, then they are congruent.
  Converse Q: If two triangles are congruent, then they have the same area.
  Determine if these statements are true or not. Justify the true statements and give a counterexample for each false statement.
• It can also happen that both the proposition and converse are false! Can you give an example?
• Here is a statement of the Baudhāyana–Pythagoras theorem.
  Let a, b, c be the sidelengths of a triangle. If the triangle is right-angled, then a² + b² = c².
  What is its converse? Is this converse true?

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