# Thread: Is this number even or odd?

1. ## Is this number even or odd?

Decide if the binomial coefficient ${100 \choose 68}$ is even or odd.

2. Originally Posted by Intsecxtanx
Decide if the binomial coefficient ${100 \choose 68}$ is even or odd.

We know (and if not prove it!) that the maximal power of a prime $p$ dividing $n!$ is given by $\sum^\infty_{k=1}\left[n/p^k\right]$ , where $[x]$ is the floor function = the greatest integer that is less than or equal $x$.

Note that the above sum is finite since $\left[n/p^r\right]=0\,\,\,for\,\,\,p^r>n$

So the maximal power of 2 dividing 100! is $50+25+12+6+3+1=97$ , and the max. power of 2 dividing 32! is $16+8+4+2+1=31$ , and the one

dividing 68 is $34+17+8+4+2+1=66$ , thus the number is odd.

Tonio