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Expression in n always divisible by k: product of consecutive integers

This type of question, asked 2 times: CDS 1 2023, CDS 2 2025. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    CDS 1 2023

    37
    Consider the following statements : 1. n3−nn^3 - n is divisible by 6. 2. n5−nn^5 - n is divisible by 5. 3. n5−5n3+4nn^5 - 5n^3 + 4n is divisible by 120. Which of the statements given above are correct ?

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  2. Question 2

    CDS 1 2023

    16
    Consider the following statements : 1. If n is a natural number, then the number n(n2+2)3\frac{n(n^2 + 2)}{3} is also a natural number. 2. If m is an odd integer, then the number m4+4m2+1116\frac{m^4 + 4m^2 + 11}{16} is an integer. Which of the statements given above is/are correct?

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  3. Question 3

    CDS 2 2025

    17
    Let x=n(n+1)(n+2)x=n(n+1)(n+2), where nn is an even natural number. Which of the following statements is/are correct? I. xx is always divisible by 48. II. x2x^2 is always divisible by 144. Select the answer using the code given below.

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