Consider the following statements :<br>1. If n is a natural number, then the number \frac{n(n^{2}+2)}{3} is also a natural number.<br>2. If m is an odd integer, then the number \frac{m^{4}+4m^{2}+11}{16} is an integer.<br>Which of the statements given above is/are correct?

Explanation

For statement 1: n(n^2+2) = n(n^2-1+3) = (n-1)n(n+1) + 3n. The product of 3 consecutive integers is divisible by 3, so the sum is divisible by 3. For statement 2: Any odd square m^2 = 8k+1. m^4+4m^2+11 = (m^2-1)(m^2-3) + 8m^2 + 8. Using modulo 16 properties for odd numbers, this expression is always perfectly divisible by 16.

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