Let S=5^a+7^b+11^c+13^d, where a, b, c and d are natural numbers. What is the number of distinct remainders of S when it is divided by 10?

Explanation

Dividing by 10 gives the unit digit. The unit digit of 5^a is always 5, and 11^c is always 1. The unit digits of 7^b and 13^d cycle through \{1, 3, 7, 9\}. The sum of any two numbers from \{1, 3, 7, 9\} can yield any even unit digit \{0, 2, 4, 6, 8\}. Thus, the unit digit of S is (5 + 1 + \text{even digit}) \pmod{10}, which results in 5 possible distinct even remainders.

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