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Remainder when a sum of two large powers is divided by a number

This type of question, asked 6 times: 25 Jul 2022, 27 Jul 2022, 28 Jul 2022, 30 Jan 2023, 1 Feb 2023, 10 Apr 2023. All 6 questions below, oldest first.

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  1. Question 1

    25 Jul 2022

    154
    The remainder when (11)1011+(1011)11(11)^{1011}+ (1011)^{11} is divided by 9 is

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  2. Question 2

    27 Jul 2022

    65
    The remainder when (2021)2022+(2022)2021(2021)^{2022}+ (2022)^{2021} is divided by 7 is

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  3. Question 3

    28 Jul 2022

    73
    The remainder when 72022+320227^{2022}+3^{2022} is divided by 5 is:

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  4. Question 4

    30 Jan 2023

    179
    50th 50^{\text{th~}} root of a number xx is 12 and 50th 50^{\text{th~}} root of another number yy is 18 . Then the remainder obtained on dividing (x+y)(x + y) by 25 is

  5. Question 5

    1 Feb 2023

    83
    The remainder, when 19200+2320019^{200}+23^{200} is divided by 49, is

  6. Question 6

    10 Apr 2023

    169
    Let the number (22)2022+(2022)22(22)^{2022}+ (2022)^{22} leave the remainder α\alpha when divided by 3 and β\beta when divided by 7. Then (α2+β2)\left( \alpha^{2}+\beta^{2} \right) is equal to

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