---
title: "Prove n even iff 7n+4 even"
subject: "Discrete Structure"
chapter: 1
key: "semester_4/discrete_structure:prove-n-even-iff-7n-plus-4-even"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/prove-n-even-iff-7n-plus-4-even
site: BCT Engineering
---

# Prove n even iff 7n+4 even

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2074 Bhadra, 5 marks
- Video: [Discrete Math - Even-Odd Proofs](https://www.youtube.com/watch?v=74pv1QykPzo) — Math Lessons by Dr. Lane
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/prove-n-even-iff-7n-plus-4-even
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

Prove that if n is a positive integer, then n is even if and only if 7n + 4 is even.

## Short answer

If n is even (n=2k), 7n+4=2(7k+2) is even. Contrapositively, if n is odd (n=2k+1), 7n+4=14k+11=2(7k+5)+1 is odd, so if 7n+4 is even then n must be even. Thus n even iff 7n+4 even.

## Answer

**To prove.** For a positive integer n: n is even ⇔ 7n + 4 is even. We prove both directions of the biconditional.

#### Part 1: if n is even, then 7n + 4 is even (direct proof)
1. Let n be even, so n = 2k for some integer k.
2. Then 7n + 4 = 7(2k) + 4 = 14k + 4 = 2(7k + 2).
3. 7k + 2 is an integer, so 7n + 4 is 2 times an integer. Hence 7n + 4 is even.

#### Part 2: if 7n + 4 is even, then n is even (proof by contraposition)
We prove the contrapositive: if n is odd, then 7n + 4 is odd.
1. Let n be odd, so n = 2k + 1 for some integer k.
2. Then 7n + 4 = 7(2k + 1) + 4 = 14k + 11 = 2(7k + 5) + 1.
3. This has the form 2m + 1, so 7n + 4 is odd.

The contrapositive is true, so the original statement is true: if 7n + 4 is even, then n is even.

#### Conclusion
Both implications hold, so n is even if and only if 7n + 4 is even. ∎

## Check yourself

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

1. What must be done to prove 'A if and only if B'?
   A. Prove only A → B
   B. Prove both A → B and B → A
   C. Prove A ∧ B
   D. Prove ¬A → ¬B only
   Answer: B. Both implications are needed. (Key point: Biconditional)

2. For n = 2k, 7n + 4 equals?
   A. 2(7k + 1)
   B. 2(7k + 2)
   C. 7(2k + 4)
   D. 14k + 5
   Answer: B. 14k + 4 = 2(7k + 2), which is even. (Key point: Direct proof)

3. To prove 7n + 4 even → n even, what is proved?
   A. If n is even, 7n + 4 is even
   B. If n is odd, 7n + 4 is odd
   C. If 7n + 4 is odd, n is even
   D. If n is odd, 7n + 4 is even
   Answer: B. The contrapositive: n odd implies 7n+4 odd. (Key point: Contraposition)

4. For n = 2k + 1, 7n + 4 equals?
   A. 14k + 4
   B. 2(7k + 5) + 1
   C. 2(7k + 2)
   D. 14k + 10
   Answer: B. 7(2k+1)+4 = 14k+11 = 2(7k+5)+1, which is odd. (Key point: Odd computation)

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