I know for you this is easy but for me is not. I give my best shot but it's no use so I need someone to teach about all this stuff.
As I try to solve this one, I come up with this answer:
Suppose $x^2-5xy-3$ is even, then $x=2a + 1$ and $y=2b$ for some integers $a,b \in\mathbb{Z}$.
Thus, $\begin{aligned}x^2-5xy-3&=(2a+1)^2-5(2a+1)(2b)-3 \\ &=(4a^2+4a+1)-20ab+10b-3 \\ &=2(2a^2+2a)-20ab+10b-2 \\ &=\;?\end{aligned}$
And I don't know what's the next step. I know there's something wrong with my procedure.
I also have plenty of other questions that need to be answered. I've already answer this equations but I can't solve it.
- If $m$ is odd and $n$ is even, then $m^2-5mn+n^2+1$ is even, where $m,n\in\mathbb{Z}$.
- If $x-y$ is even, then $x^2+3xy-5$ is odd, where $m,n \in\mathbb{Z}$.
- Let $a,b \in\mathbb{Z}$. If $2b^2-3ab+1$ is even, then $2a-b$ is odd.
- Let $m,n \in\mathbb{Z}$. Prove that if $m^2+1$ is even, then $2n+m$ is even.
Even though I'm not good in math, I know in the future I will be good in math by practicing and with your help.
This not a assignment, I'm practicing solving problems like this to be good in math.