Consider the following statements :

1. If n is a natural number, then the number \(\frac{n\left(n^2+2\right)}{3}\) is also a natural number.

2. If m is an odd integer, then the number \(\frac{\mathrm{m}^4+4 \mathrm{~m}^2+11}{16}\) is an integer. 

Which of the statements given above is/are correct ? 

Calculation:

Statement 1:

Let n be 1

So, \(\frac{n\left(n^2+2\right)}{3} = ​​\frac{1\left(1^2+2\right)}{3} \)

⇒ \( ​​\frac{1\left(1+2\right)}{3} \)

⇒ \(​​\frac{3}{3} \)

⇒ 1

So, statement 1 is correct.

Statement 2: 

Let m be 3

So, \(\frac{\mathrm{m}^4+4 \mathrm{~m}^2+11}{16} = \frac{\mathrm{3}^4+4 (3)^2+11}{16} \)

⇒ \(\frac{81+4 \times9+11}{16} \)

⇒ \(\frac{81+36+11}{16} \)

⇒ \(\frac{128}{16} \)

⇒ 8

So, statement 2 is correct.

∴ The required answer is Option 3.