Let $a_1, a_2, ..., a_n$ be natural numbers and $d$ the gcd of all of them. Using Bezout lemma in number theory we can find integers $k_1, k_2,..., k_n$ such that $a_1k_1 + a_2k_2 + ... + a_nk_n = d$.
Are there always integers $k_1,k_2,..., k_n$ such that $a_1k_1 + a_2k_2 + ... + a_nk_n = d$ and $k_2\mid k_3\mid k_4\mid \dots\mid k_n$ ?
I am trying to solve a linear algebra problem and in the middle of my solution I faced this stronger form of the lemma and I am not able to solve it the question without this.