I have a trouble understanding p.7 of the following article: http://www.edb.gov.hk/attachment/en/curriculum-development/kla/ma/IMO/Nov20155-4online.pdf
which says the folllowing:
By the same reasoning as before (the chinese remainder theorem), we know that there exists an integer $a$ such that $n=m_1m_2\cdots m_9+a, 2m_1m_2\cdots m_9 +a, 3m_1m_2\cdots m_9 +a, \cdots$ all make $n+1, n+2, \cdots ,n+9$ composite.
I didn't understand why such $a$ exist. Why the same argument still holds for an "infinite system of congruence" as above? Thanks for help.