 Number Theory

Linear Diophantine Equations - One Equation In a certain basketball game, you can only make 2 point shots and 3 point shots. If you scored exactly 20 points, how many of the following scenarios are possible? 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? Jimmy has the high score, 920, in his favorite video game. For each win, he gets 50 points, but for each loss, he loses 36 points. What is the minimum number of games that Jimmy could have played? 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! Gummy bears come in packs of 6 and 9. Is there a combination of these packs which gives a total of exactly 100 gummy bears?

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