RO-TST 2012 Problem 1

Let $n_1,\ldots,n_k$ be positive integers, and define $d_1=1$ and $d_i=\frac{(n_1,\ldots,n_{i-1})}{(n_1,\ldots,n_{i})}$ , for $i\in \{2,\ldots,k\}$ , where $(m_1,\ldots,m_{\ell})$ denotes the greatest common divisor of the integers $m_1,\ldots,m_{\ell}$ . Prove that the sums $$\sum_{i=1}^k a_in_i$$ with $a_i\in\{1,\ldots,d_i\}$ for $i\in\{1,\ldots,k\}$ are mutually distinct $\mod n_1$ .