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What is a Pythagorean triplet?
A Pythagorean triplet is a set of three positive integers (a, b, c) that satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
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Finding Pythagorean triplets:
To find Pythagorean triplets for a given number, you can use the following formula:
- For any two positive integers m and n, where m > n, the Pythagorean triplet can be generated as follows:
a = m^2 - n^2
b = 2mn
c = m^2 + n^2
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Steps to find Pythagorean triplets:
1. Choose two positive integers m and n, where m > n.
2. Substitute these values into the formulas for a, b, and c as mentioned above.
3. Calculate the values of a, b, and c to obtain the Pythagorean triplet.
4. Verify that the triplet satisfies the Pythagorean theorem by checking if a^2 + b^2 = c^2.
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Example:
Let's say we want to find a Pythagorean triplet for the number 5:
1. Choose m = 2 and n = 1 (m > n).
2. Substitute these values into the formulas:
a = 2^2 - 1^2 = 3
b = 2 * 1 * 2 = 4
c = 2^2 + 1^2 = 5
3. The Pythagorean triplet for 5 is (3, 4, 5).
4. Verify: 3^2 + 4^2 = 9 + 16 = 25 = 5^2, which satisfies the Pythagorean theorem.
By following these steps and formulas, you can find Pythagorean triplets for any given number.