Question - Solving a System of Equations with Integer Solutions
Solution:
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.
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