Find two positive values, m and n, that satisfy the following conditions:
The value of m > n > 0.
Both m and n are coprime.
Either m or n is an even integer.
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?
To find two positive values m and n that satisfy the given conditions, we first need to understand Euclid's formula for generating Pythagorean triples. Euclid's formula states that for any positive integers m and n with m > n, a primitive Pythagorean triple can be generated using the following equations:
a = m^2 - n^2
b = 2mn
c = m^2 + n^2
Here, a, b, and c represent the sides of the right-angled triangle, with c being the hypotenuse.
Now, let's find two positive values m and n that satisfy the given conditions:
1. The value of m > n > 0: This condition tells us that m must be greater than n, and both values must be positive.
Let's choose m = 3 and n = 2 to satisfy this condition.
2. Both m and n are coprime: For two numbers to be coprime, their greatest common divisor (GCD) must be 1.
The GCD of 3 and 2 is 1, so this condition is satisfied.
3. Either m or n is an even integer: Let's take n as the even integer in this case. Since 2 is an even integer, this condition is satisfied.
Using these values of m and n, we can substitute them into Euclid's formula to generate a primitive Pythagorean triple:
a = m^2 - n^2 = 3^2 - 2^2 = 9 - 4 = 5
b = 2mn = 2 * 3 * 2 = 12
c = m^2 + n^2 = 3^2 + 2^2 = 9 + 4 = 13
Therefore, the Pythagorean triple generated by these values is (5, 12, 13).
Now, let's consider what would happen if the values of m and n were both even or both odd:
If both m and n were even, then they would not be coprime since they would have a common divisor of 2. Therefore, a primitive Pythagorean triple cannot be generated in this case.
If both m and n were odd, then they would still be coprime, but neither would be even. However, if we examine Euclid's formula, we can see that when both m and n are odd, the resulting values of a and b will also be odd. This violates the condition that either m or n should be an even integer. So, even in this case, a primitive Pythagorean triple cannot be generated.
In conclusion, a primitive Pythagorean triple can only be generated when one of the values m or n is even.