Find two positive values, m and n, that satisfy the following conditions:
a.The value of m > n > 0.
b.Both m and n are coprime.
C.Either m or n is an even integer.
2. Use Euclid's formula to generate a primitive Pythagorean triple using these values of m and n.
3. What do you think will happen if the values of m and n are both even or both odd?
4.Can a primitive Pythagorean triple still be generated using values that are both even or both odd?
1. One possible pair of values that satisfy the given conditions is m = 4 and n = 2. Both m and n are positive, m > n, and they are coprime since their greatest common divisor is 1. Additionally, n is an even integer.
2. To generate a primitive Pythagorean triple using Euclid's formula, the formula is:
a = m² - n²
b = 2mn
c = m² + n²
Using m = 4 and n = 2:
a = 4² - 2² = 12
b = 2(4)(2) = 16
c = 4² + 2² = 20
So the primitive Pythagorean triple is (12, 16, 20).
3. If both m and n are even, then the numbers will not be coprime since they will share a common factor of 2. Similarly, if both m and n are odd, they will also not be coprime since they will share a common factor of 1.
4. A primitive Pythagorean triple cannot be generated using values that are both even or both odd. In a primitive Pythagorean triple, the values of a, b, and c must be coprime, and if both m and n are even or both odd, they will have a common factor, making them not coprime.