Explanation:

- Let's analyze the statements using the condition \( a + b + c = abc \).

- Statement 1: The arithmetic mean of \( a, b, \) and \( c \) is \(\frac{a+b+c}{3}\). Given \( a + b + c = abc \), a specific solution is \( a = 1, b = 2, \) and \( c = 3 \). For these values, the arithmetic mean is \(\frac{6}{3} = 2\), a natural number. So, statement 1 is correct.

- Statement 2: The harmonic mean of \( a, b, \) and \( c \) is \(\frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}\). Using \( a = 1, b = 2, \) and \( c = 3 \), it simplifies to \(\frac{3}{\frac{11}{6}}\), which equals \(\frac{18}{11}\). Since \(\frac{18}{11}\) is between 1 and 2, statement 2 is also correct.

- The correct option is:

Option 3: Both 1 and 2