2023 · Q17
It is given that 5 does not divide n - 1, n and n + 1, where n is a positive integer. Which one of the following is correct?
Elementary MathematicsNumber System
- A5 divides (n² + 1)
- B5 divides (n² - 1)
- C5 divides (n² + n)
- D5 divides (n²- n)
Explanation
Since 5 divides none of n − 1, n, n + 1, we must have n ≡ 2 or 3 (mod 5). Then n² ≡ 4 (mod 5) in both cases, so n² + 1 ≡ 0 (mod 5). The alternatives fail for n = 2 (e.g. n² − 1 = 3), so 5 divides n² + 1 (A).