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Sum or difference of powers of one base: factor out the smallest power to find a divisor

This type of question, asked 5 times: CDS 2 2016, CDS 1 2018, CDS 1 2020, CDS 1 2022, CDS 1 2026. All 6 questions below, oldest first.

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  1. Question 1

    CDS 2 2016

    30
    2122+462+842+464+21302^{122} + 4^{62} + 8^{42} + 4^{64} + 2^{130} is divisible by which one of the following integers ?

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  2. Question 2

    CDS 1 2018

    1
    517+518+519+5205^{17} + 5^{18} + 5^{19} + 5^{20} is divisible by

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  3. Question 3

    CDS 1 2020

    10
    Which one of the following is the largest divisor of 3x+3x+1+3x+23^{x} + 3^{x+1} + 3^{x+2}, if xx is any natural number ?

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  4. Question 4

    CDS 1 2022

    4
    461+462+463+4644^{61} + 4^{62} + 4^{63} + 4^{64} is divisible by

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  5. Question 5

    CDS 1 2022

    21
    275+31327^5 + 3^{13} is divisible by

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  6. Question 6

    CDS 1 2026

    21
    2727−940−37927^{27} - 9^{40} - 3^{79} is divisible by how many natural numbers less than 10 ?

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