Given Information:
- p(x) is a real polynomial of least degree.
- p(x) has a local maximum at x = 1 and a local minimum at x = 3.
- p(1) = 6 and p(3) = 2.

Local Maximum and Minimum:
- A local maximum occurs at a point where the function reaches its highest value in a small interval around that point.
- Similarly, a local minimum occurs at a point where the function reaches its lowest value in a small interval around that point.

Properties of Local Maximum and Minimum:
- At a local maximum or minimum, the derivative of the function is zero or undefined.
- At a local maximum, the second derivative of the function is negative.
- At a local minimum, the second derivative of the function is positive.

Derivative of p(x):
- Since p(x) is a polynomial, it is differentiable for all values of x.
- The derivative of p(x) gives the slope of the tangent line to the graph of p(x) at any point.
- At the local maximum and minimum points, the derivative of p(x) is zero.

Using the given information:
- The derivative of p(x) is zero at x = 1 and x = 3.
- Let's assume the degree of p(x) is n.
- Since p(x) is a polynomial of least degree, it means p(x) is a polynomial of degree n.
- The derivative of p(x) is a polynomial of degree (n-1).
- Therefore, p'(x) has at least two roots, x = 1 and x = 3.
- Since p'(x) has degree (n-1), it can have at most (n-1) roots.

Inferring the degree of p(x):
- Since p'(x) has at least two roots (x = 1 and x = 3), the degree of p'(x) is at least 2.
- Therefore, the degree of p(x) is at least 3.

Form of p(x):
- The form of p(x) can be written as p(x) = a(x - 1)(x - 3)(x - c) where a is a constant and c is a real number.
- This form satisfies the conditions of having a local maximum at x = 1 and a local minimum at x = 3.

Using p(1) = 6 and p(3) = 2:
- Plug in x = 1 and x = 3 into the form of p(x).
- p(1) = a(1 - 1)(1 - 3)(1 - c) = 0
- p(3) = a(3 - 1)(3 - 3)(3 - c) = 0
- Since p(1) = 6 and p(3) = 2, the only possibility is that (1 - c) = -2 and (3 - c) = 2.
- Solving the equations, we get c = 3.

Calculating p(0):
- Plug in x = 0 into the form of p(x).
- p(0) = a(0