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Distinct last digits of a sum of powers over all exponents

This type of question, asked 2 times: CDS 1 2024, CDS 2 2024. All 3 questions below, oldest first.

Try each one first, then open its answer.

  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 1 2024

    18
    If n is a natural number, then what is the sum of all distinct remainders of 4n+6n+9n+11n4^n + 6^n + 9^n + 11^n when divided by 10 for various values of n ?

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  3. Question 3

    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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