Let \(A \) be an 16 by 16 matrix, with entries \( a_{i,j} = \gcd(i, j )\) for \(1 \leq i, j \leq 16 \). Calculate the last 3 digits of \(\det (A) \).

**Details and assumptions**

\( \det(A)\) refers to the determinant of matrix \(A\).

\(\gcd(i,j) \) refers to the greatest common divisor of \(i\) and \(j\).

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