Solving a System of Equations with Integer Solutions
Sure, let's solve the questions provided in the image one by one.
1. Solve in \( \mathbb{Z} \) the system \( (x^2 - y^2 = 5440, \operatorname{gcd}(x;y) = 8) \)
We know that \( x^2 - y^2 = (x+y)(x-y) \), and since the greatest common divisor of \( x \) and \( y \) is 8, we can write \( x = 8m \) and \( y = 8n \), where \( m \) and \( n \) are integers that are coprime (since if \( m \) and \( n \) had a common factor, then \( x \) and \( y \) would have a greater common divisor than 8).
Now our equation becomes:
\[ (8m + 8n)(8m - 8n) = 5440 \]
\[ 64(m + n)(m - n) = 5440 \]
\[ (m + n)(m - n) = \frac{5440}{64} \]
\[ (m + n)(m - n) = 85 \]
The factors of 85 are 1, 5, 17, and 85. Since \( m \) and \( n \) are integers and \( m > n \), the pairs \((m, n)\) that satisfy \( (m+n)(m-n) = 85 \) are \((43, 42)\) and \((9, 8)\).
So, plugging these values back into \( x = 8m \) and \( y = 8n \), the solutions are:
For \((m, n) = (43, 42)\):
\[ x = 8 \cdot 43 = 344 \]
\[ y = 8 \cdot 42 = 336 \]
For \((m, n) = (9, 8)\):
\[ x = 8 \cdot 9 = 72 \]
\[ y = 8 \cdot 8 = 64 \]
Thus, our solutions in \( \mathbb{Z} \) for the system are \((x, y) = (344, 336)\) and \((x, y) = (72, 64)\).
If you need assistance with the other problems in the image, please let me know, and I can help you with those as well.
