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Difference of squares x^2 - y^2 = k: use factor pairs (count or find solutions)

This type of question, asked 3 times: CDS 1 2019, CDS 2 2019, CDS 2 2024. All 5 questions below, oldest first.

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  1. Question 1

    CDS 1 2019

    96
    What is the sum of all integer values of nn for which n2+19n+92n^2 + 19n + 92 is a perfect square ?

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  2. Question 2

    CDS 1 2019

    90
    How many pairs of natural numbers are there such that the difference of their squares is 35 ?

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  3. Question 3

    CDS 2 2019

    20
    Suppose n is a positive integer such that (n2+48)(n^2 + 48) is a perfect square. What is the number of such n ?

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  4. Question 4

    CDS 2 2024

    96
    The number 199 can be written as m2−n2m^2 - n^2, where mm, nn are natural numbers (m>n)(m > n). What is the value of mnmn ?

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  5. Question 5

    CDS 2 2024

    2
    The difference of the square of two natural numbers mm and nn (m>n)(m > n) is 72. How many pairs of natural numbers will satisfy ?

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