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!

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