I can't do this poof of being even I know these things If $x$ is divided by $11$ it remainder of $7$. If $x$ is divided by $13$ it remainder of $7$. If $x$ is divide by $3$ it remainder $3$. $x$ is less than $500$. ...--prophetes.ai

I can't do this poof of being even I know these things If $x$ is divided by $11$ it remainder of $7$. If $x$ is divided by $13$ it remainder of $7$. If $x$ is divide by $3$ it remainder $3$. $x$ is less than $500$. I am trying to prove that $x$ even ...

**Hint** $\ $ If $\,x\,$ is a solution so too is $\,x + 3\cdot 11\cdot 13,$ and they have _opposite_ parity.

* * *

Maybe you seek to prove that the _least nonegative_ solution is even. Then

coprime $\,11,13\mid x-7\,\Rightarrow\, 11\cdot 13= 143\mid x-7\ $ so $\,x = 7 + 143k$.

Thus ${\rm mod}\ 3\\!:\,\ 3 \equiv x \equiv 7+143k\equiv 1+2k\ $ so $\,2k\equiv 2,\,$ so $\,k\equiv 1\ $ so $\, k = 1+3n$

Thus $\,x = 7+143(1+3n) = 150+ 429n$