Squares of consecutive odd integers — Aptitude Question
Squares of consecutive odd integers: The difference between the squares of two consecutive odd integers is always… Answer and worked solution on CodeKairo.
- Topic: Number System
- Section: Quantitative Aptitude
- Difficulty: Medium
- Time target: 75 seconds
Question
The difference between the squares of two consecutive odd integers is always divisible by:
Options
- 3
- 6
- 7
- 8
Show the answer and the worked solution
Correct answer: D. 8
Worked solution
- Let the integers be n and n + 2 with n odd.
- (n + 2)² − n² = 4n + 4 = 4(n + 1).
- Since n is odd, n + 1 is even, so 4(n + 1) is a multiple of 8.
- Check: 5² − 3² = 16, 7² − 5² = 24, 9² − 7² = 32 — all multiples of 8. ✓
Answer: 8.
The idea
Algebra first, then test on the smallest case. The parity of n is what upgrades the obvious factor of 4 to a factor of 8, and it is exactly the step candidates skip.
Shortcut. Test the smallest pair: 3² − 1² = 8, and of the options only 8 divides 8.
Common trap. 6 — testing only 5 and 7 gives 49 − 25 = 24, which 6 divides; the pair 3 and 5 gives 16, which it does not.