# 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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