Question

In a 5th graders response

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.
Use Euclid's formula to generate a primitive Pythagorean triple using these values of m and n.
Euclid's Formula:
a = m2- n2
b = 2mn
c = m2 + n2

Answer 1

To find two positive values, \( m \) and \( n \), that satisfy the conditions:
1. \( m > n > 0 \)
2. Both \( m \) and \( n \) are coprime (which means their greatest common divisor is 1)
3. Either \( m \) or \( n \) is even

Let's choose \( m = 4 \) and \( n = 1 \).

1. **Check the conditions**:
- \( m > n \): Yes, \( 4 > 1 \).
- \( n > 0 \): Yes, \( 1 > 0 \).
- Coprime: \( \text{gcd}(4, 1) = 1\), so they are coprime.
- Either \( m \) or \( n \) is even: Yes, \( m = 4 \) is even.

Now that we have \( m = 4 \) and \( n = 1 \), we can use Euclid’s formula to generate a primitive Pythagorean triple:

Using the formulas:
- \( a = m^2 - n^2 \)
- \( b = 2mn \)
- \( c = m^2 + n^2 \)

Now we can calculate \( a \), \( b \), and \( c \).

**Step 1: Calculate \( a \)**
\[
a = m^2 - n^2 = 4^2 - 1^2 = 16 - 1 = 15
\]

**Step 2: Calculate \( b \)**
\[
b = 2mn = 2 \times 4 \times 1 = 8
\]

**Step 3: Calculate \( c \)**
\[
c = m^2 + n^2 = 4^2 + 1^2 = 16 + 1 = 17
\]

The primitive Pythagorean triple generated is \( (a, b, c) = (15, 8, 17) \).

So the final answer is:
- \( m = 4 \)
- \( n = 1 \)
- The Pythagorean triple is \( (15, 8, 17) \).