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Sum of consecutive powers of a number: which number divides it

The same question, asked 2 times: 18 Jul 2023 S1*, 26 Sep 2024 S3. Every wording below, oldest first.

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  1. Question 1

    18 Jul 2023 S1

    66
    Which of the following numbers will completely divide 412+413+414+4154^{12} + 4^{13} + 4^{14} + 4^{15}?

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  2. Question 2

    26 Sep 2024 S3

    67
    715+716+7177^{15} + 7^{16} + 7^{17} is divisible by which of the following given numbers?

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