2025 · Q17
Let x = n(n+1)(n+2), where n is an even natural number. Which of the following statements is/are correct?
- Ix is always divisible by 48.
- IIx^2 is always divisible by 144.
Elementary MathematicsNumber System
- AI only
- BII only
- CBoth I and II
- DNeither I nor II
Explanation
n even, x=n(n+1)(n+2). I: n=2 gives x=24, not divisible by 48, so I is false. II: write n=2m, x=4m(m+1)(2m+1), so x^2=16m^2(m+1)^2(2m+1)^2. Since 3 always divides m(m+1)(2m+1) (check residues mod 3), 9 divides the square, and 16*9=144 divides x^2. Only II holds. The n=2 counterexample kills I instantly.