Is it possible to write a Pythagorean triple in which 2 of the numbers are even and one is odd?
To determine if it is possible to write a Pythagorean triple where two numbers are even and one is odd, we need to understand the properties of Pythagorean triples.
A Pythagorean triple consists of three positive integers (a, b, c) that satisfy the equation a^2 + b^2 = c^2. This equation represents the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
Let's consider the parity (evenness or oddness) of perfect squares. The square of an even number is always even, and the square of an odd number is always odd. Additionally, the sum of two even numbers is even, the sum of two odd numbers is even, and the sum of an odd and an even number is odd.
Now, let's assume that we have a Pythagorean triple with two even numbers (a and b) and one odd number (c). We can consider the three possible cases:
1. If both a and b are even, their squares (a^2 and b^2) will be even. However, the sum of two even numbers (a^2 + b^2) will also be even, which means that c^2 (and therefore c) must also be even. This contradicts the initial condition that c is odd, so a Pythagorean triple with two even numbers and one odd number is not possible.
2. If both b and c are even, their squares (b^2 and c^2) will be even. Consequently, a^2 (and therefore a) must also be even, as their sum (a^2 + b^2) must yield an even value. This contradicts the initial condition that one number is odd, so a Pythagorean triple with two even numbers and one odd number is not possible.
3. If both a and c are even, their squares (a^2 and c^2) will be even. In this case, b^2 (and therefore b) must be odd, as their sum (a^2 + b^2) must yield an even value. However, the square of an odd number is odd, so it is not possible to have an odd value for c^2. Therefore, a Pythagorean triple with two even numbers and one odd number is not possible.
Based on the analysis, it is not possible to write a Pythagorean triple where two of the numbers are even and one is odd.