Question #41237

This question constructs a 'Lethal Collision' between Linear Algebra and Number Theory.

The primary trap lies in the summation definition of the matrix entries, which students often attempt to simplify using row operations or telescopic sums, leading to a massive time sink.

The intended path requires recognizing the matrix as a product $A D A^T$, where $A$ is a lower triangular Pascal matrix with determinant 1.

Once reduced to the determinant of the diagonal matrix $D$, the problem shifts to Number Theory.

A secondary trap occurs when applying Legendre's formula: students frequently miscalculate the exponent of a prime in a product of factorials by incorrectly swapping summations or miscounting the modulo remainders for $\lfloor 2k/3^j \rfloor$.