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?
Step-by-step solution
Idea: Count factors pair by pair: a pair of two different numbers gives 2 factors, a repeated pair gives only 1.
- Every factor that is not in a repeated pair has a partner different from itself, so these factors come in twos: an even number of them, say 2k.½ mark
- Each repeated pair gives only one factor. The two repeated pairs (f, f) and (g, g) give 2 factors: f and g.½ mark
- Total = 2k + 2 = 2(k + 1), an even number.½ mark
- That is why (ii) alone cannot give (i). In fact this case never happens: f × f = g × g with f, g positive means f = g, so a number has at most one repeated pair.½ mark
Answer to write in the exam
Factors not in repeated pairs come in pairs of two different numbers: 2k factors
Repeated pairs (f, f), (g, g) add 2 factors: f, g
Total = 2k + 2 = 2(k + 1)
∴ The number of factors would be even.
Common mistakes that cost marks
- Counting a repeated pair as two factors. (f, f) gives only one factor, f.
- Answering ‘odd’ because the number is a square. The question asks about the (impossible) case of two repeated pairs.
How this can come in the exam
A number has factor–partner pairs (1, 400), (2, 200), (4, 100), (5, 80), (8, 50), (10, 40), (16, 25), (20, 20). How many factors does it have?
- 8
- 15
- 16
- 17
Show answer
(B) 15
Seven pairs of different numbers give 14 factors, and (20, 20) gives one more: 15.
Explain why no positive integer can have two different factor–partner pairs (f, f) and (g, g).
Show answer
Both pairs multiply to the same number, so f × f = g × g (1 mark). If f < g then f × f < g × g, and if f > g then f × f > g × g; so f = g and the pairs are the same (1 mark).Try one yourself
225 has factor–partner pairs (1, 225), (3, 75), (5, 45), (9, 25), (15, 15). How many factors does it have, and why is the number odd?
Show answer
Four pairs of different numbers give 8 factors and (15, 15) gives one more: 9 factors. It is odd because exactly one pair repeats.
More questions like this
- 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. - If two lines are parallel, then the corresponding angles formed by a transversal are equal.