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a^n - b^n is divisible by a - b, and a^n + b^n by a + b when n is odd

This type of question, asked 7 times: CDS 2 2016, CDS 2 2019, CDS 1 2021, CDS 1 2022, CDS 1 2023, CDS 2 2024, CDS 2 2025. All 10 questions below, oldest first.

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  1. Question 1

    CDS 2 2016

    97
    What is the remainder when 135+145+155+16513^5 + 14^5 + 15^5 + 16^5 is divided by 29 ?

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  2. Question 2

    CDS 2 2019

    50
    The expression 52n−23n5^{2n} - 2^{3n} has a factor

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  3. Question 3

    CDS 2 2019

    1
    What is the remainder when (1729+1929)(17^{29} + 19^{29}) is divided by 18 ?

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  4. Question 4

    CDS 1 2021

    44
    If nn is any natural number, then 52n−15^{2n} - 1 is always divisible by how many natural numbers?

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  5. Question 5

    CDS 1 2021

    6
    What is the remainder when 2727−152727^{27} - 15^{27} is divided by 6?

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  6. Question 6

    CDS 1 2022

    2
    4143+434341^{43} + 43^{43} is divisible by

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  7. Question 7

    CDS 1 2023

    32
    The number 9730−143097^{30} - 14^{30} is divisible by :

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  8. Question 8

    CDS 2 2024

    64
    A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : Is 327n+173n327^n + 173^n divisible by 500 ? Statement-I : nn is odd natural number Statement-II : nn is a positive integer Which one of the following is correct in respect of the above Question and the Statements ?

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  9. Question 9

    CDS 2 2024

    76
    Consider the following statements : I. 61 divides 107100−76100107^{100} - 76^{100} II. 100 divides 675+33567^5 + 33^5 Which of the statements given above is/are correct ?

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  10. Question 10

    CDS 2 2025

    11
    What is the remainder when (1725+1925)(17^{25}+19^{25}) is divided by 18?

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