How Many Numbers?

Number Theory Level pending

Find the number of integers \(N\) lying between \(2\) and \(100\) (inclusive) such that for all integers \(m\) such that \(\dfrac{N}{3} \leq m \leq \dfrac{N}{2},\) \(\dbinom{m}{N-2m}\) is a multiple of \(m.\)

Details and assumptions

  • \(N=2\) is considered a solution, since the only integer \(m\) such that \(\dfrac{2}{3} \leq m \leq \dfrac{2}{2}\) is \(1,\) and \(\dbinom{2}{2 - 2 \cdot 1}\) is a multiple of \(1.\)

  • This problem is not original.

@Calvin sir: The intended expression is \(\dbinom{m}{N-2m}.\) This is valid since \(m \geq N - 2m \geq 0\) from the condition \(\dfrac{N}{3} \leq m \leq \dfrac{N}{2}.\) Roger Lu has commented why \(\dbinom{N}{N-2m}\) doesn't work. Please correct me if I am wrong.


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