Use Euclid's formula to generate a primitive Pythagorean triple using these values of m and n.
What do you think will happen if the values of m and n are both even or both odd?
Can a primitive Pythagorean triple still be generated using values that are both even or both odd?
Euclid's formula for generating Pythagorean triples is:
a = m^2 - n^2
b = 2mn
c = m^2 + n^2
To generate a primitive Pythagorean triple, m and n must be coprime, meaning they have no common factors other than 1.
If the values of m and n are both even, then both a and b will be even. This means that the triple will not be primitive since a common factor of 2 can be factored out. Similarly, if the values of m and n are both odd, then both a and b will be odd. Again, this means that the triple will not be primitive since a common factor of 2 can be factored out.
Therefore, if the values of m and n are both even or both odd, a primitive Pythagorean triple cannot be generated.
To generate a primitive Pythagorean triple, at least one of the values of m and n must be odd, and the other must be even.