Question
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?
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?
Answers
Answered by
GPT-4o mini
To find two positive values \( m \) and \( n \) that satisfy the conditions \( m > n > 0 \), coprime, and with either \( m \) or \( n \) being even, we can select \( m = 3 \) and \( n = 2 \).
1. **Check Conditions**:
- \( m = 3 \), \( n = 2 \) such that \( 3 > 2 > 0 \).
- Coprime: \( \gcd(3, 2) = 1 \) (since 3 and 2 have no common divisors).
- Either \( m \) or \( n \) is even: \( n = 2 \) is even.
Now, let's use these values to generate a primitive Pythagorean triple using Euclid's formula, which states that for two coprime integers \( m \) and \( n \) with \( m > n \), the primitive Pythagorean triple can be generated by:
\[
a = m^2 - n^2, \quad b = 2mn, \quad c = m^2 + n^2
\]
2. **Calculate the values of \( a \), \( b \), and \( c \)**:
\[
a = 3^2 - 2^2 = 9 - 4 = 5
\]
\[
b = 2 \cdot 3 \cdot 2 = 12
\]
\[
c = 3^2 + 2^2 = 9 + 4 = 13
\]
Thus, the primitive Pythagorean triple generated is \( (5, 12, 13) \).
### Analysis of Both Odd or Both Even
If both \( m \) and \( n \) are both odd or both even, the generated values will not yield a primitive Pythagorean triple.
- **Both Values Odd**:
- If \( m \) and \( n \) are both odd, then:
- \( m^2 - n^2 \) is even,
- \( 2mn \) is even,
- \( m^2 + n^2 \) is even.
- All three sides would have a common factor of 2, hence cannot form a primitive Pythagorean triple.
- **Both Values Even**:
- If \( m \) and \( n \) are both even, then:
- Similarly, the values generated would also be divisible by 2 (they will all be even).
- This violates the definition of primitive triples, which require the values to have no common factor other than 1.
### Conclusion
Thus, you cannot generate a primitive Pythagorean triple using values that are both odd or both even. A primitive Pythagorean triple can only be generated when one of the integers is even and the other is odd.
1. **Check Conditions**:
- \( m = 3 \), \( n = 2 \) such that \( 3 > 2 > 0 \).
- Coprime: \( \gcd(3, 2) = 1 \) (since 3 and 2 have no common divisors).
- Either \( m \) or \( n \) is even: \( n = 2 \) is even.
Now, let's use these values to generate a primitive Pythagorean triple using Euclid's formula, which states that for two coprime integers \( m \) and \( n \) with \( m > n \), the primitive Pythagorean triple can be generated by:
\[
a = m^2 - n^2, \quad b = 2mn, \quad c = m^2 + n^2
\]
2. **Calculate the values of \( a \), \( b \), and \( c \)**:
\[
a = 3^2 - 2^2 = 9 - 4 = 5
\]
\[
b = 2 \cdot 3 \cdot 2 = 12
\]
\[
c = 3^2 + 2^2 = 9 + 4 = 13
\]
Thus, the primitive Pythagorean triple generated is \( (5, 12, 13) \).
### Analysis of Both Odd or Both Even
If both \( m \) and \( n \) are both odd or both even, the generated values will not yield a primitive Pythagorean triple.
- **Both Values Odd**:
- If \( m \) and \( n \) are both odd, then:
- \( m^2 - n^2 \) is even,
- \( 2mn \) is even,
- \( m^2 + n^2 \) is even.
- All three sides would have a common factor of 2, hence cannot form a primitive Pythagorean triple.
- **Both Values Even**:
- If \( m \) and \( n \) are both even, then:
- Similarly, the values generated would also be divisible by 2 (they will all be even).
- This violates the definition of primitive triples, which require the values to have no common factor other than 1.
### Conclusion
Thus, you cannot generate a primitive Pythagorean triple using values that are both odd or both even. A primitive Pythagorean triple can only be generated when one of the integers is even and the other is odd.