Prove that, for all $ \in \mathbb{Z}$, $x + p$ and $(x + p)(x − p)$ leave the same remainder when divided by $2$.
∈ = belong
Z = set of the integers
p = 3
My tentative approach is given below:
$x = y \ ( \text{mod } 2)$
$x + 3 = y \ ( \text{mod } 2)$
$x = 2(x + 3) + y$
$x_1 = y \ ( \text{mod } 2)$
$(x + 3) (x + 3) = y \ ( \text{mod } 2)$
$x_1 = 2[(x + 3) (x - 3)] + y$
$x_1 = 2[x² - 3x + 3x - 9] + y$
Can someone help me with this problem? I can't solve it. Thanks!!
Case x + p is even. Clearly (x + p)(x - p) is even
Case x + p is odd.
Assume x - p is even. Then 2x = x + p + x - p is odd.
Thus x - p is odd and accordingly (x + p)(x - p) is odd.