MathNet 2011 Question 24

Numbers $a_1, a_2, a_3, a_4, a_5$ and $b_1, b_2, b_3, b_4, b_5$ are permutations of $1, 2, 3, 4, 5$. Prove that among five numbers $a_1b_1, a_2b_2, a_3b_3, a_4b_4, a_5b_5$ at least two have the same remainders modulo $5$.