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Number of distinct remainders mod 10 of a sum of four powers over all exponents

The same question, asked 2 times: CDS 1 2024*, CDS 2 2024. Every wording below, oldest first.

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  1. Question 1

    CDS 1 2024

    17
    If a, b, c, d are natural numbers, then how many possible remainders are there when 1a+2b+3c+4d1^a + 2^b + 3^c + 4^d is divided by 10 ?

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  2. Question 2

    CDS 2 2024

    99
    Let S=5a+7b+11c+13dS = 5^a + 7^b + 11^c + 13^d, where aa, bb, cc and dd are natural numbers. What is the number of distinct remainders of SS when it is divided by 10 ?

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