Remainder when a large power (or sum of powers) is divided by a small number: cycle or a = +-1

This type of question, asked 8 times: CDS 1 2017, CDS 2 2019, CDS 2 2020, CDS 1 2021, CDS 2 2021, CDS 1 2023, CDS 2 2024, CDS 1 2025. All 12 questions below, oldest first.

Try each one first, then open its answer.

Question 1
CDS 1 2017
8 $(4444)^{4444}$
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Question 2
CDS 2 2019
33 $3^{521}$
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Question 3
CDS 2 2020
7 $1^5 + 2^5 + 3^5 + 4^5 + 5^5$
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Question 4
CDS 2 2020
11 $17^{2020}$
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Question 5
CDS 1 2021
2 $2^{1000000}$
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Question 6
CDS 2 2021
4 $37^{1000}$
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Question 7
CDS 1 2023
3 $65^{99}$
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Question 8
CDS 2 2024
4 $111^{222} + 222^{333} + 333^{444}$
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Question 9
CDS 2 2024
73 $3^{10}$
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Question 10
CDS 2 2024
26 $x\,(0 \le x \le 8)$
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Question 11
CDS 2 2024
25 $3^{255}$
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Question 12
CDS 1 2025
5 $p$
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Remainder when a large power (or sum of powers) is divided by a small number: cycle or a = +-1

Question 1

Question 2

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Question 9

Question 10

Question 11

Question 12