Non-negative fractions summing to $1$

Question

Let $ d_1,\ldots, d_n \ge 2 $ be pairwise relatively prime. Are there any $ c_1,\ldots,c_n \in \mathbb{Z}_{\ge 0} $ with $ c_i \le d_i-1 $ for all $ i=1,\ldots,n $, such that $\displaystyle \sum_{i=1}^n \dfrac{c_i}{d_i} =1 $?

I see the answer is negative for all of the smaller cases I considered. But I have no proof. This is a conjecture given me by my friend, but probably this can be done with elementary tools, so I am posting it here instead of MO. I hope someone will help me.

Answer 1

Hint: Multiply both sides of the equation $\sum_{i=1}^n \frac{c_i}{d_i}=1$ by the product $d_2\cdots d_n$.

Answer 2

If it is the case $d_1\cdots d_n=c_1d_2\cdots d_n+$ a multiple of $d_1$, that is, $d_1$ divides $c_1d_2\cdots d_n$. Since $d_1$ is coprime with $d_i$, $2\leq i\leq n$, then $d_1$ divides $c_1$, but this is impossible.