Ta co: $(1+x)^{n}=C^{0}_{n}+C^{1}_{n}.x+C^{2}_{n}.x^{2}+...+C^{k}_{n}.x^{k}+C^{n}_{n}.x^{n}$Cho x=-1 ta duoc:$0=C^{0}_{n}-C^{1}_{n}+C^{2}_{n}-C^{3}_{n}+...+C^{2k}_{n}-C^{2k+1}_{n}+...+(-1)^{n}C^{n}_{n}$
$\Rightarrow$$C^{0}_{n}+C^{2}_{n}+C^{4}_{n}+...=C^{1}_{n}+C^{3}_{n}+C^{5}_{n}+...$ (1)
Cho x=1 ta co:$2^{n}=C^{0}_{n}+C^{1}_{n}+C^{2}_{n}+C^{3}_{n}+....C^{2k}_{n}+C^{2k+1}_{n}+...+C^{n-1}_{n}+C^{n}_{n}$
Ap dung (1) ta co : $2^{n}=2(C^{0}_{n}+C^{2}_{n}+C^{4}_{n}+...)$
$\Rightarrow$$\frac{2^{n}}{2}=C^{0}_{n}+C^{2}_{n}+C^{4}_{n}+...$hay$2^{n-1}=C^{0}_{n}+C^{2}_{n}+C^{4}_{n}+...$(2)
Tu (1) va(2)$\Rightarrow$dpcm