Least natural $n$ such that $\,n \bmod 4\, =\, 3 + (n \bmod 34)$ [duplicate]
Solution 1:
You want to find the least number $n$ such that \begin{eqnarray*} n\%34&\equiv& r,\\ n\%4&\equiv& r+3, \end{eqnarray*} Of course by definition \begin{eqnarray*} 0\leq&n\%34&\leq33,\\ 0\leq&n\%4&\leq3, \end{eqnarray*} from which it follows that $r=0$. Then $n=34x$ and $n=4y+3$ for some integers $x$ and $y$. But the former implies that $n$ is even, whereas the latter implies that $n$ is odd, a contradiction. Hence no such number exists.