Pythagorean Triplets

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 hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In other words, for a Pythagorean triplet (a, b, c), a² + b² = c².

Finding a Pythagorean Triplet of 82

To find a Pythagorean triplet of 82, we need to find two numbers (a and b) whose squares add up to 82² = 6724.

Let's start by assuming a = 1 and b = 2. We can then calculate c using the Pythagorean theorem:

a² + b² = c²
1² + 2² = c²
1 + 4 = c²
5 = c²
c = √5

However, c is not an integer, so this does not form a Pythagorean triplet.

We can continue this process by increasing the values of a and b and calculating c until we find a Pythagorean triplet.

Let's try a = 1 and b = 3:

a² + b² = c²
1² + 3² = c²
1 + 9 = c²
10 = c²
c = √10

Again, c is not an integer, so this is not a Pythagorean triplet.

Finding a Pythagorean Triplet of 82 (continued)

We can continue this process until we find a Pythagorean triplet. However, in this case, we will not find a Pythagorean triplet of 82 because 82 is not a perfect square.

The Pythagorean theorem only holds for right-angled triangles, and the lengths of the sides are given by positive integers. Since 82 is not a perfect square, we cannot find a Pythagorean triplet for it.

Hence, there is no Pythagorean triplet of 82.

Summary:
- A Pythagorean triplet consists of three positive integers (a, b, c) that satisfy the Pythagorean theorem: a² + b² = c².
- To find a Pythagorean triplet of 82, we need to find two numbers (a and b) whose squares add up to 82² = 6724.
- By trying different values of a and b and calculating c, we can determine if a Pythagorean triplet exists.
- However, since 82 is not a perfect square, we cannot find a Pythagorean triplet for it.
- Therefore, there is no Pythagorean triplet of 82.

For finding Pythagorean triplet - we have formula. (2m,m²+1,m²-1)
so our smallest number take as 82 so first applying 2m we got 2m=82,m=41.
we got m right now we are finding our second number of triplet (m²+1),m=41,41+1=42.
now,find third one (m²-1) ,41-1=40.
so,here we made our whole phythagorean triplet(82,42,40)