There are 404 numbers between 1 and 2020 inclusive that can be the side of the hypotenuse given that three sides of the right triangle have integral lengths which form an arithmetic sequence. This can be obtained by forming the arithmetic sequence, equating by Pythagoras theorem and finding numbers divisible by the integral.
Let the arithmetic sequence be (a - d), a, (a+d)
Using Pythagoras theorem,
(a - d)² + a² = (a+d)²
a² -2ad + d² + a² = a² + 2ad + d²
2a² - 2ad + d² = a² + 2ad + d²
2a² - a² + d² - d² = 2ad + 2ad
a² = 4ad
a = 4d
Thus hypotenuse will be (a + d) = 4d + d = 5d
Since the value of hypotenuse is 5d, the total numbers divisible by 5 between 1 and 2020 will be the number of possible sides of the hypotenuse.
There are 2020 numbers between 1 and 2020 inclusive
The numbers divisible by 5 = 2020/5 = 404
Hence there are 404 numbers between 1 and 2020 inclusive that can be the side of the hypotenuse given that three sides of the right triangle have integral lengths which form an arithmetic sequence.
Learn more about arithmetic sequence here:
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