Using mathematical induction to prove a divisibility

Question

I am having trouble to prove that if $a\mid b_1, a\mid b_2, \dots, a\mid b_n$, then $a \mid (b_1 + b_2 + \cdots + b_n)$.

Answer 1

By hypothesis, $b_1 = a \times c_1$, $b_2 = a \times c_2$, $\dots$, $b_n = a \times c_n$. Then $b_1 + b_2 + \dots + b_n = a \times c_1 + a \times c_2 + \dots + a \times c_n = a(c_1 + c_2 + \dots + c_n)$. Hence $a$ divides the sum $b_1 + b_2 + \dots + b_n$. Hope this helps.