It is not possible to construct a magic square that is both additive and multiplicative, even if the sum and the product are different. To understand why this is true, we can use the distributive property of multiplication over addition.

Suppose we have a 5x5 magic square that is both additive and multiplicative, with a magic sum S and a magic product P. We can then write the following equations:

x₁ + x₂ + x₃ + x₄ + x₅ = S
x₆ + x₇ + x₈ + x₉ + x₁₀ = S
x₁₁ + x₁₂ + x₁₃ + x₁₄ + x₁₅ = S
x₁₆ + x₁₇ + x₁₈ + x₁₉ + x₂₀ = S
x₂₁ + x₂₂ + x₂₃ + x₂₄ + x₂₅ = S

x₁ * x₂ * x₃ * x₄ * x₅ = P
x₆ * x₇ * x₈ * x₉ * x₁₀ = P
x₁₁ * x₁₂ * x₁₃ * x₁₄ * x₁₅ = P
x₁₆ * x₁₇ * x₁₈ * x₁₉ * x₂₀ = P
x₂₁ * x₂₂ * x₂₃ * x₂₄ * x₂₅ = P

Now, if we multiply all the equations in the first list, we get:

(x₁ + x₂ + x₃ + x₄ + x₅) * (x₆ + x₇ + x₈ + x₉ + x₁₀) * (x₁₁ + x₁₂ + x₁₃ + x₁₄ + x₁₅) * (x₁₆ + x₁₇ + x₁₈ + x₁₉ + x₂₀) * (x₂₁ + x₂₂ + x₂₃ + x₂₄ + x₂₅) = S⁵

And if we multiply all the equations in the second list, we get:

(x₁ * x₂ * x₃ * x₄ * x₅) * (x₆ * x₇ * x₈ * x₉ * x₁₀) * (x₁₁ * x₁₂ * x₁₃ * x₁₄ * x₁₅) * (x₁₆ * x₁₇ * x₁₈ * x₁₉ * x₂₀) * (x₂₁ * x₂₂ * x₂₃ * x₂₄ * x₂₅) = P⁵

However, since the magic square is both additive and multiplicative, we can equate the two expressions above:

S⁵ = P⁵

But this contradicts Fermat's Last Theorem, which states that there are no integer solutions to the equation xⁿ + yⁿ = zⁿ when n is greater than 2. Therefore, it is not possible to construct a magic square that is both additive and multiplicative, even if the sum and the product are different.