Supposing Newton's law of gravitation for gravitation forces `F_(1)` and `F_(2)` between two masses `m_(1)` and `m_(2)` at positions `r_(1)` and `r_(2)` read
where `M_(0)` is a constant of dimension of mass, `r_(12) = r_(1) - r_(2)` and `n` is a number. In such a case,

Options

A. the acceleration due to gravity on the earth will be different for different objects
B. none of the three laws of Kepler will be valid
C. only the third law will become invalid
D. for `n` negative, an object lighter than water will sink in water

Explanation

To solve the problem, we need to analyze the modified gravitational force equation given in the question and understand its implications on the acceleration due to gravity and the behavior of objects in a gravitational field.

### Step-by-Step Solution:

1. **Understanding the Gravitational Force Equation**:
The gravitational force between two masses \( m_1 \) and \( m_2 \) is given by:
\[ F_1 = -F_2 = -\frac{G \cdot m_0^2 \cdot m_1 \cdot m_2}{r_{12}^3} \]
where \( r_{12} = r_1 - r_2 \) is the position vector between the two masses, and \( n \) is a number that modifies the force.

2. **Expressing Acceleration due to Gravity**:
The acceleration due to gravity \( g \) can be expressed as:
\[ g = \frac{F}{m} \]
Substituting the expression for force \( F \):
\[ g = -\frac{G \cdot m_0^2 \cdot m_1 \cdot m_2}{r_{12}^3 \cdot m} \]

3. **Analyzing the Dependence on Position**:
From the expression for \( g \), we can see that \( g \) depends on the position vector \( r_{12} \) and is inversely proportional to \( r_{12}^2 \). This means that the value of \( g \) can vary depending on the position of the masses involved.

4. **Implications for Different Objects**:
Since \( g \) is dependent on the position vector, if two objects are at different positions in the gravitational field, they will experience different values of \( g \). Therefore, the acceleration due to gravity on Earth will not be constant for different objects.

5. **Evaluating the Options**:
- **Option A**: None of the laws of Kepler will be valid. This is incorrect because the central force nature of gravity remains intact.
- **Option B**: The first and second laws of Kepler remain valid, but the third law will not hold due to the variable \( g \).
- **Option C**: For \( n < 0 \), an object lighter than water will sink in water. This is a tricky statement but can be true under the modified gravitational conditions.

### Conclusion:
The correct interpretation of the problem leads us to conclude that the acceleration due to gravity will indeed vary for different objects based on their position in the gravitational field. Therefore, the answer to the question is that the acceleration due to gravity on Earth will be different for different objects.