Prove that the number $ 9^{(a_1 + a_2)(a_2 + a_3)(a_3 + a_4)...(a_{98} + a_{99})(a_{99} + a_1)}$ − $1$ is divisible by $10$, for any choice of positive integers $a_1, a_2, a_3, . . . , a_{99}$.
Source: OMK 2018 Bongsu, Section B Problem 2
Tags: Proof, number theory, contests
Prove that the number $ 9^{(a_1 + a_2)(a_2 + a_3)(a_3 + a_4)...(a_{98} + a_{99})(a_{99} + a_1)}$ − $1$ is divisible by $10$, for any choice of positive integers $a_1, a_2, a_3, . . . , a_{99}$.