If n and n + 3 have no factors in common, then n is not a multiple of 3.
Step-by-step solution
To find: Frame the converse of this proposition. Then, determine if each of the two statements is true or not. Justify the true statements and give a counterexample for each false statement.
Idea: Any common factor of n and n + 3 also divides their difference, 3. So the only possible common factors are 1 and 3, and 3 is a common factor exactly when 3 divides n.
- Converse: If n is not a multiple of 3, then n and n + 3 have no factors in common.½ mark
- 1 divides every number, so ‘no factors in common’ means no common factor other than 1.
- Proposition — true. Suppose n were a multiple of 3: n = 3k. Then n + 3 = 3k + 3 = 3(k + 1) is also a multiple of 3, so 3 would be a common factor. That is not allowed, so n is not a multiple of 3.1 mark
- Converse. Let d be a common factor of n and n + 3. Then d divides their difference (n + 3) − n = 3, so d = 1 or d = 3.½ mark
- d = 3 would mean 3 divides n, which is ruled out. So d = 1: no common factor other than 1. The converse is true.½ mark
- So the proposition and its converse are both true.½ mark
Check: n = 4: 4 and 7 share only 1 ✓. n = 10: 10 and 13 share only 1 ✓. n = 6 (a multiple of 3): 6 and 9 share 3 ✓.
Answer to write in the exam
Converse: If n is not a multiple of 3, then n and n + 3 have no factors in common.
(‘No factors in common’ = no common factor other than 1.)
Proposition: if n = 3k, then n + 3 = 3(k + 1), so 3 is a common factor — contradiction. ∴ 3 ∤ n. True.
Converse: common factor d divides (n + 3) − n = 3 ⇒ d = 1 or 3
3 ∤ n ⇒ d ≠ 3 ⇒ d = 1. True.
∴ Both the proposition and its converse are true.
Common mistakes that cost marks
- Saying the proposition can never apply because 1 is always a common factor. ‘No factors in common’ ignores the factor 1.
- Checking a few values of n and calling it a proof. Use the fact that a common factor divides the difference, 3.
How this can come in the exam
What is the HCF of n and n + 3 when n = 25?
- 1
- 3
- 5
- 25
Show answer
(A) 1
25 is not a multiple of 3, so 25 and 28 have no common factor other than 1.
Show that n and n + 2 have no common factor other than 1 whenever n is odd.
Show answer
A common factor divides (n + 2) − n = 2, so it is 1 or 2 (1 mark). n is odd, so 2 does not divide n; the only common factor is 1 (1 mark).Try one yourself
For which of n = 9, 10, 11, 12 do n and n + 3 have a common factor other than 1?
Show answer
n = 9 (9 and 12 share 3) and n = 12 (12 and 15 share 3) — exactly the multiples of 3.
More questions like this
- There are no known ‘neat’ expressions that generate only primes! Find counterexamples to the following claims.
- Find counterexamples to the following statements.
- Consider the statement: ‘If a number is divisible by 8, then it is divisible by both 2 and 4’.
- Recall that a shortcut to check whether a given number is divisible by 3 is to add the digits of the number and check if the sum is a multiple of 3. Express the relationship between ‘a number is divisible by 3’ and ‘sum of the digits is a multiple of 3’ using ‘If-then’ sentences.
- We have identified different types of quadrilaterals — squares, rectangles, parallelograms, rhombi, kites and trapezia. One can identify more types (e.g., we could create a category of quadrilaterals that have equal-length opposite sides).
Suppose we have identified a category of quadrilaterals called Q, and we have to construct a quadrilateral of this type. For this, we are to use two thin sticks, put them together as diagonals so that the quadrilateral obtained by joining their endpoints is of type Q (see the figure).