Divisible expression
Can you show this algebraic expression is divisible by 4?
Age
14 to 16
Challenge level
Problem
Show that $(1+x+y)^2-(1-x-y)^2$ is divisible by $4$ for all integer values of $x$ and $y$.
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
$\begin{align}(1+x+y)^2&=1+x+y+x+x^2+xy+y+xy+y^2\\
&=1+2x+2y+2xy+x^2+y^2\\
\ \\
(1-x-y)^2&=1-x-y-x+x^2+xy-y+xy+y^2\\
&=1-2x-2y+2xy+x^2+y^2\\
\ \\
\text{difference}&=4x+4y\\
&=4(x+y)\\
&=4\times\text{(an integer), which is a multiple of 4.}\end{align}$