(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)?
Step-by-step solution
Idea: Factors in pairs of two different numbers always give an even count; each repeated pair adds just one factor. So whether the total is odd depends on how many repeated pairs there are, which (ii) does not say.
- (ii) says: some factor–partner pair repeats a number. It says nothing about how many such pairs there are.½ mark
- Pairs of two different numbers always give an even number of factors. Each repeated pair adds only one factor. So one repeated pair gives an odd total, but two repeated pairs would give an even total.½ mark
- So (ii) on its own does not force (i): the step from (ii) to (i) has a gap.½ mark
- The gap is closed by noting that a number has at most one repeated pair (f × f = g × g forces f = g). With that extra fact, (ii) does lead to (i).½ mark
Answer to write in the exam
(ii) gives at least one repeated pair, not the number of such pairs.
One repeated pair: factors = even + 1 = odd; two repeated pairs: even + 2 = even.
(ii) ⇒ (i) only with the extra fact ‘at most one repeated pair’ (f2 = g2 ⇒ f = g).
∴ Not necessarily: (ii) alone does not imply (i).
Common mistakes that cost marks
- Answering ‘yes, obviously’. (ii) by itself does not count the repeated pairs.
- Thinking ‘implies’ works backwards automatically: (i) ⇒ (ii) does not give (ii) ⇒ (i).
How this can come in the exam
Assertion (A): ‘x = 2′ implies ‘x2 = 4′.
Reason (R): ‘x2 = 4′ implies ‘x = 2′.
- Both Assertion (A) and Reason (R) are true and R is the correct explanation of A.
- Both Assertion (A) and Reason (R) are true but R is not the correct explanation of A.
- Assertion (A) is true but Reason (R) is false.
- Assertion (A) is false but Reason (R) is true.
Show answer
(C) Assertion (A) is true but Reason (R) is false.
A is true: 22 = 4. R is false: x = −2 also gives x2 = 4.
S: ‘n is a multiple of 9′. T: ‘n is a multiple of 3′. Which is correct?
- S implies T, but T does not imply S
- T implies S, but S does not imply T
- S and T imply each other
- Neither implies the other
Show answer
(A) S implies T, but T does not imply S
9k = 3(3k), so S ⇒ T. But 3 is a multiple of 3 and not of 9, so T does not imply S.
Try one yourself
A: ‘n ends in 5′. B: ‘n is divisible by 5′. Does A imply B? Does B imply A?
Show answer
A ⇒ B: yes, a number ending in 5 is divisible by 5. B ⇒ A: no, 10 is divisible by 5 but ends in 0.
More questions like this
- 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? - 1. Identify real-life examples in which a proposition is true but not the converse. Give nice counterexamples!
2. Give more examples from geometry as well as number theory in which a proposition is true but not its converse. Give appropriate counterexamples.