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A parity problem
11-24-2010, 04:24 PM
Post: #1
A parity problem
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.
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06-03-2016, 08:54 AM
Post: #2
RE: A parity problem
Proof $\quad\because\; 2(a_0+a_2+a_4+\cdots +a_{50})$
$\qquad\qquad\;\; = (1+x+x^2)^{25}|_{x=1}+ (1+x+x^2)^{25}|_{x=-1}=3^{25}+1\equiv (-1)^{25}+1\equiv 0\pmod{4}$

$\qquad\quad\;\;\therefore\; a_0+a_2+a_4+\cdots +a_{50}\equiv 0\pmod{2}.\quad\square$
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