---
title: "Converse, inverse, contrapositive of prime statement"
subject: "Discrete Structure"
chapter: 1
key: "semester_4/discrete_structure:converse-inverse-contrapositive-prime"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/converse-inverse-contrapositive-prime
site: BCT Engineering
---

# Converse, inverse, contrapositive of prime statement

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2075 Bhadra, 3 marks; 2070 Magh, 3 marks (asked 2 times)
- Video: [Converse, Inverse, Contrapositive of Implication || Propositions || DMS || Discrete Mathematics](https://www.youtube.com/watch?v=9mVRLkKoi7Q) — Sudhakar Atchala
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/converse-inverse-contrapositive-prime
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

State the converse, contrapositive and inverse of the statement, "A positive integer is prime only if it has no divisors other than 1 and itself".

## Short answer

Original: If a positive integer is prime then it has no divisors other than 1 and itself (p→q). Converse (q→p): If it has no such divisors then it is prime. Inverse (¬p→¬q): If not prime then it has another divisor. Contrapositive (¬q→¬p): If it has another divisor then it is not prime.

## Answer

Let p: "a positive integer is prime" and q: "it has no divisors other than 1 and itself".

"p only if q" means p → q. So the given statement is: **If a positive integer is prime, then it has no divisors other than 1 and itself.**

| Form | Symbolic | Statement |
|---|---|---|
| Converse | q → p | If a positive integer has no divisors other than 1 and itself, then it is prime. |
| Inverse | ¬p → ¬q | If a positive integer is not prime, then it has a divisor other than 1 and itself. |
| Contrapositive | ¬q → ¬p | If a positive integer has a divisor other than 1 and itself, then it is not prime. |

The contrapositive is logically equivalent to the original statement. The converse and the inverse are equivalent to each other, but not to the original statement in general.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is 'p only if q' written symbolically?
   A. q → p
   B. p → q
   C. p ↔ q
   D. ¬p → q
   Answer: B. 'p only if q' means p → q: if p then q. (Key point: 'Only if' translation)

2. What is the contrapositive of 'If a number is prime, it has no divisors other than 1 and itself'?
   A. If it has no other divisors, then it is prime
   B. If it is not prime, it has a divisor other than 1 and itself
   C. If it has a divisor other than 1 and itself, it is not prime
   D. If it is prime, it has a divisor other than 1 and itself
   Answer: C. The contrapositive of p → q is ¬q → ¬p. (Key point: Contrapositive)

3. Which pair is logically equivalent?
   A. Original and converse
   B. Original and inverse
   C. Converse and inverse
   D. Original and contrapositive only, converse and inverse are not equivalent to each other
   Answer: C. The converse (q → p) and inverse (¬p → ¬q) are contrapositives of each other, so they are equivalent; the original is equivalent only to its contrapositive. (Key point: Equivalences)

## Related in Discrete Structure

- [Contrapositive and inverse: snow statement](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Acontrapositive-inverse-snow-home) — Discrete Structure, chapter 1
- [Converse etc. of temple statement](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Aconverse-contrapositive-temple-saturday) — Discrete Structure, chapter 1
- [Converse etc. of beach statement](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Aconverse-contrapositive-inverse-beach) — Discrete Structure, chapter 1

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