If n is divisible by 60, then it is divisible by both 5 and 12.
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: Unlike 4 and 6, the numbers 5 and 12 share no prime factor. So if 5 divides 12k, the 5 must come from k, which makes n a multiple of 60.
- Converse: If n is divisible by both 5 and 12, then it is divisible by 60.½ mark
- Proposition — true. If n = 60k, then n = 5 × 12k and n = 12 × 5k, so n is divisible by 5 and by 12.1 mark
- Converse — true. Since 12 divides n, write n = 12k. Since 5 divides n, the prime 5 appears in the prime factorisation of 12k. 12 = 2 × 2 × 3 has no factor 5, so 5 must divide k.1 mark
- So k = 5m and n = 12 × 5m = 60m: n is divisible by 60. Both statements are true.½ mark
Check: Multiples of 12: 12, 24, 36, 48, 60, 72, … — the first one ending in 0 or 5 (divisible by 5) is 60, and the next is 120 = 2 × 60.
Answer to write in the exam
Converse: If n is divisible by both 5 and 12, then it is divisible by 60.
Proposition: n = 60k = 5(12k) = 12(5k). True.
Converse: n = 12k; 5 | 12k and 5 ∤ 12 (12 = 2 × 2 × 3), 5 prime ⇒ 5 | k
k = 5m ⇒ n = 60m. True.
∴ Both the proposition and its converse are true.
Common mistakes that cost marks
- Using the idea from 24 = 4 × 6 here and calling the converse false. 5 and 12 have no common factor, so the converse holds.
- ‘Proving’ the converse by checking 60, 120 and 180 only. A general argument is needed.
How this can come in the exam
For which pair (p, q) is ‘if n is divisible by both p and q, then n is divisible by pq‘ true for every positive integer n?
- (4, 10)
- (6, 9)
- (7, 9)
- (8, 12)
Show answer
(C) (7, 9)
7 and 9 have no common factor other than 1. The others fail: 20 (for 4, 10), 18 (for 6, 9), 24 (for 8, 12).
Assertion (A): Every number divisible by both 3 and 8 is divisible by 24.
Reason (R): 3 and 8 have no common factor other than 1.
- 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
(A) Both Assertion (A) and Reason (R) are true and R is the correct explanation of A.
Because 3 and 8 share no factor, n = 8k with 3 | 8k forces 3 | k, so 24 | n. R explains A.
Try one yourself
Is ‘If n is divisible by both 4 and 9, then n is divisible by 36′ true? Give a reason.
Show answer
True. n = 9k; 4 divides 9k and 9 = 3 × 3 has no factor 2, so 4 divides k. Then n = 36m.
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