Number Theory
# Linear Diophantine Equations

Consider all ordered pairs of integers $(x,y)$ such that $\frac{5}{x}+\frac{7}{y} = \frac{12}{xy}.$

The smallest positive integer value of $x$ in these ordered pairs is $1,$ since $x=y=1$ satisfies the equation. What is the second smallest positive integer value of $x$ in these ordered pairs?

If you have infinite pennies ($0.01), nickels ($0.05), and dimes ($0.10), in how many different ways can you make change for $1.00?

**Hint:** Don't try to list all of the possibilities!