Distributive factorials

Number Theory Level 4

How many ordered pairs of positive integers \(1 \leq k \leq n \leq 50\) are there, such that \(k\) divides \(n\), and

\[ \left(\frac{n}{k}\right)! = \frac{n!}{k!} \quad ? \]

Details and assumptions

For an ordered pair of integers \((a,b)\), the order of the integers matter. The ordered pair \((1, 2)\) is different from the ordered pair \((2,1) \).


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