Consider the following argument.
Each factor of a number has a ‘partner’ factor such that their product yields the given number, e.g., 5 is a factor of 35, and 5 × 7 = 35. Here, 7 is the partner factor of 5, and vice-versa. Let us focus on factor-partner pairs of numbers, e.g., (1, 12), (2, 6), (3, 4) are the pairs for the number 12.
If a number has an odd number of factors, there must be a factor-partner pair in which the same number repeats (e.g., the partner factor of 5 in 25). If not, the given number will have an even number of factors since each factor can be paired with its factor pair.
Thus, if a number has an odd number of factors, then it is a perfect square.
What does this argument prove? Statement P or Q?
Step-by-step solution
Idea: An argument proves ‘if X then Y’ when it assumes X and arrives at Y. So look at what it starts from and where it ends.
- Where does the argument start? ‘If a number has an odd number of factors …’ — it assumes the number has an odd number of factors.½ mark
- Where does it end? ‘… then it is a perfect square.’½ mark
- So it shows: odd number of factors ⇒ perfect square. That is the statement ‘If n has an odd number of factors, then it is a perfect square’.½ mark
- This is Statement Q. It does not prove P, which goes the other way (perfect square ⇒ odd number of factors); P needs its own argument.½ mark
Answer to write in the exam
Assumes: the number has an odd number of factors.
Concludes: the number is a perfect square.
Odd number of factors ⇒ perfect square
∴ The argument proves Statement Q (not P).
Common mistakes that cost marks
- Answering P because the argument talks about perfect squares. What matters is the direction: what is assumed and what is concluded.
- Thinking that proving Q also proves P. A statement and its converse need separate proofs.
How this can come in the exam
An argument begins ‘Suppose a triangle has three equal angles’ and ends ‘… so the triangle has three equal sides’. Which statement does it prove?
- If a triangle has three equal sides, then it has three equal angles.
- If a triangle has three equal angles, then it has three equal sides.
- Both of these statements.
- Neither statement.
Show answer
(B) If a triangle has three equal angles, then it has three equal sides.
It assumes ‘three equal angles’ and concludes ‘three equal sides’, so it proves ‘if three equal angles, then three equal sides’ only.
Riya argues: ‘Let n = 4k. Then n = 2(2k), so n is even.’ Which proposition has she proved: ‘If n is even, then n is a multiple of 4′ or ‘If n is a multiple of 4, then n is even’? Is the other one true?
Show answer
She assumed n is a multiple of 4 and concluded n is even, so she proved ‘If n is a multiple of 4, then n is even’ (1 mark). The other is false: 6 is even but not a multiple of 4 (1 mark).Try one yourself
An argument starts with ‘Let n be odd’ and ends with ‘so n2 is odd’. Write the proposition it proves, and write its converse.
Show answer
It proves ‘If n is odd, then n2 is odd’. Converse: ‘If n2 is odd, then n is odd’ (also true, but it needs its own argument).
More questions like this
- This only proves Q. It doesn’t prove that every square number has an odd number of factors (Proposition P).
Is P true? - (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?