Question

If $p$ is a prime number, show that the coefficients of the terms of $(1+x{)}^{p-1}$ are alternately greater and less by unity than some multiple of $p$.

Answer 1

Let the co efficients be denoted by $C_0,C_1,C_2,.........C_r,,..$

then $C_r=\frac{(p-1)(p-2)(p-3)...(p-r)}{r!}$

$=\frac{M\left(p\right)+(-1)r!}{r!}=\frac{M\left(p\right)}{r!}+{(-1)}^r$

Now, $C_r$ is a positive integer;hence

$\frac{M\left(p\right)}{r!}+{\left(1\right)}^r$ must be a positive integer, and since p is a prime number it must be a multiple of p; therefore

$C_r=M\left(p\right)+{(-1)}^r$