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?
To satisfy the given conditions, let's try the values m = 4 and n = 3.
a. The value of m (4) is greater than n (3), and both are positive.
b. Both m and n are coprime since their greatest common divisor is 1.
c. 4 is an even integer.
Using Euclid's formula, we can generate a primitive Pythagorean triple:
a = m^2 - n^2 = 4^2 - 3^2 = 16 - 9 = 7
b = 2mn = 2 * 4 * 3 = 24
c = m^2 + n^2 = 4^2 + 3^2 = 16 + 9 = 25
So, a = 7, b = 24, and c = 25 form a primitive Pythagorean triple.
If the values of m and n are both even or both odd, it is not possible to satisfy condition c since in that case, neither m nor n would be an even integer.
No, a primitive Pythagorean triple cannot be generated using values that are both even or both odd.