1. **Problem Statement:** We need to compute the exact value of $k = \gcd(d, x^2 - 1)$, where $d$ and $x$ are integers. 2. **Understanding the problem:** The greatest common divisor (gcd) of two integers is the largest positive integer that divides both numbers without leaving a remainder. 3. **Factorize the expression:** Note that $x^2 - 1$ is a difference of squares, so it can be factored as: $$x^2 - 1 = (x - 1)(x + 1)$$ 4. **Interpreting $d$:** Since the problem does not specify $d$, we consider $d$ as an integer that might share common factors with $x^2 - 1$. 5. **Compute $k$:** The gcd of $d$ and $x^2 - 1$ is the gcd of $d$ and $(x - 1)(x + 1)$: $$k = \gcd(d, (x - 1)(x + 1))$$ 6. **Exact value depends on $d$ and $x$:** Without specific values for $d$ and $x$, the exact numeric value of $k$ cannot be determined. 7. **If $d$ divides $x^2 - 1$ exactly:** Then $k = d$. 8. **If $d$ and $x^2 - 1$ are coprime:** Then $k = 1$. **Final answer:** The exact value of $k$ depends on the values of $d$ and $x$. It is given by: $$k = \gcd\bigl(d, (x - 1)(x + 1)\bigr)$$