+26367 Find the solutions to \(x^2 - y^2 = 100\) in positive integers.
Divisors of 100:
1 | 2 | 4 | 5 | 10 | 20 | 25 | 50 | 100 (9 divisors)
Formula:
\(\begin{array}{|rcll|} \hline \mathbf{4xy} &=& \mathbf{ (x+y)^2-(x-y)^2 } \\\\ xy &=& \left( \dfrac{x+y}{2} \right)^2- \left( \dfrac{x-y}{2} \right)^2 \\ \hline \end{array}\)
\(\begin{array}{|l | c | c | c | } \hline x*y & \dfrac{x+y}{2} & \dfrac{x+y}{2} & \text{solutions} \\ \hline 100 = 1\times 100 & \dfrac{101}{2}\ \text{odd} & \dfrac{100}{2} \\ \hline 100 = 2\times 50 & \dfrac{52}{2}=26 & \dfrac{48}{2}=24 & 26^2-24^2 = 10^2 \\ \hline 100 = 4\times 25 & \dfrac{29}{2}\ \text{odd} & \dfrac{21}{2} \ \text{odd} \\ \hline 100 = 5\times 20 & \dfrac{25}{2}\ \text{odd} & \dfrac{15}{2}\ \text{odd} \\ \hline 100 = 10\times 10 & \dfrac{20}{2} & \dfrac{0}{2}=0\ \text{not positive integer} \\ \hline \end{array}\)
