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# number theory

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Find the solutions to x^2 - y^2 = 100 in positive integers.

May 17, 2020

#1
+21959
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There are two positive answers:  (6, 8)  and  (8, 6).

May 17, 2020
#2
+25648
+2

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:

$$\begin{array}{|rcll|} \hline \mathbf{4xy} &=& \mathbf{ (x+y)^2-(x-y)^2 } \\\\ xy &=& \left( \dfrac{x+y}{2} \right)^2- \left( \dfrac{x-y}{2} \right)^2 \\ \hline \end{array}$$

$$\begin{array}{|l | c | c | c | } \hline x*y & \dfrac{x+y}{2} & \dfrac{x+y}{2} & \text{solutions} \\ \hline 100 = 1\times 100 & \dfrac{101}{2}\ \text{odd} & \dfrac{100}{2} \\ \hline 100 = 2\times 50 & \dfrac{52}{2}=26 & \dfrac{48}{2}=24 & 26^2-24^2 = 10^2 \\ \hline 100 = 4\times 25 & \dfrac{29}{2}\ \text{odd} & \dfrac{21}{2} \ \text{odd} \\ \hline 100 = 5\times 20 & \dfrac{25}{2}\ \text{odd} & \dfrac{15}{2}\ \text{odd} \\ \hline 100 = 10\times 10 & \dfrac{20}{2} & \dfrac{0}{2}=0\ \text{not positive integer} \\ \hline \end{array}$$

May 18, 2020