A parity problem - Printable Version +- OMath! (http://math.elinkage.net) +-- Forum: Math Forums (/forumdisplay.php?fid=4) +--- Forum: Number Theory (/forumdisplay.php?fid=7) +--- Thread: A parity problem (/showthread.php?tid=112) A parity problem - elim - 11-24-2010 04:24 PM If $a_0,a_1,\dots,a_{50}$ are the coefficients of the polynomial $\left(1 + x + x^2\right)^{25}$ show that $a_0+a_2+a_4+\cdots+a_{50}$ is even. RE: A parity problem - elim - 06-03-2016 08:54 AM Proof $\quad\because\; 2(a_0+a_2+a_4+\cdots +a_{50})$ $\qquad\qquad\;\; = (1+x+x^2)^{25}|_{x=1}+ (1+x+x^2)^{25}|_{x=-1}=3^{25}+1\equiv (-1)^{25}+1\equiv 0\pmod{4}$ $\qquad\quad\;\;\therefore\; a_0+a_2+a_4+\cdots +a_{50}\equiv 0\pmod{2}.\quad\square$