Prove exists consecutive integers $s,s+1,\dots,s+i$ such that $m_{i}|s+i$, if $m_{0},\dots,m_{r}$ are positive integers coprime

Question

If $m_{0},m_{1},\dots,m_{r}$ are positive integers with $(m_{i},m_{j})=1$ for $i\neq j$, prove that exists consecutive integers $s,s+1,\dots,s+i$ such that $m_{i}|s+i$ for all $1\leq i \leq r$.

I really don't know how to approach to this problem. Any tips?

The bounds on your index $1\le i\le r$ does not make clear whether it must be the case that $m_0\mid s$. — Keith Backman, Oct 04 '18 at 02:00

score 1 · Accepted Answer · answered Oct 04 '18 at 01:58

We want the solution to the following system. Since $\gcd(m_i,m_j)=1$ so we can apply Chinese Remainder Theorem to guarantee the existence of a solution. \begin{align*} s & \equiv 0 \pmod{m_0}\\ s & \equiv -1 \pmod{m_1}\\ s & \equiv -2 \pmod{m_2}\\ \vdots & \equiv \vdots\\ s & \equiv -r \pmod{m_r} \end{align*}

Prove exists consecutive integers $s,s+1,\dots,s+i$ such that $m_{i}|s+i$, if $m_{0},\dots,m_{r}$ are positive integers coprime

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