2023 · Q16
Consider the following statements :
- 1If n is a natural number, then the number (n(n²+2))/(3) is also a natural number.2. If m is an odd integer, then the number (m⁴+4m²+11)/(16) is an integer.
Which of the statements given above is/are correct?
Elementary MathematicsNumber System
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
For statement 1, n(n² + 2)/3 is integral because either 3 divides n or n² ≡ 1 (mod 3), making n² + 2 ≡ 0 (mod 3). For statement 2, with m odd write m² = 8k + 1; then m⁴ + 4m² + 11 = 16(4k² + 3k + 1), divisible by 16. Both statements are correct (C).