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Right triangle with sides consecutive integers or in terms of x: solve with Pythagoras

This type of question, asked 3 times: CDS 2 2019, CDS 2 2020, CDS 1 2021. All 3 questions below, oldest first.

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  1. Question 1

    CDS 2 2019

    77
    Three consecutive integers form the lengths of a right-angled triangle. How many sets of such three consecutive integers is/are possible ?

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  2. Question 2

    CDS 2 2020

    80
    The lengths of the sides of a right-angled triangle are consecutive even integers (in cm). What is the product of these integers ?

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  3. Question 3

    CDS 1 2021

    81
    The sides of a right-angled triangle are in the ratio x:(x−1):(x−18)x : (x - 1) : (x - 18). What is the perimeter of the triangle?

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