Please help finding positive integers less than $1000$ which satisfy the constraint: $x=7k , x=4l+2 , x=3m+1$ Please help finding positive integers less than $1000$ which satisfy the constraint: $x=7k , x=4l+2 , x=3m+1$

We want to find integers $x$ that satisfy$$\begin{align}x&=7k\\x&=4l+2\\x&=3m+1\end{align}$$
Substitute $x=7k$ into the second equation and obtain
$$7k = 4l+2\\\implies 8k-k=4l+2\\\implies -k=4\heartsuit+2\\\implies k = 4n+2 $$
So we have $k=4n+2$, where $n$ is an integer. Then
$$x=7(4n+2) = 28n+14$$
For $x$ to satisfy the last equation, we must have
$$28n+14=3m+1\\\implies n+27n+14=3m+1\\\implies n = 3\spadesuit -1\\\implies n = 3s+2$$
This yields
$$x=28(3s+2)+14 = 84s+70$$
All in all, you want to find the integers $x=84s+70$ such that
$$\color{blue}{0\lt 84s+70\lt 1000}$$
