Number Theory
# Linear Diophantine Equations

What is the positive four-digit integer for which:

- the first (in other words, the left-most) digit is one-third the second digit,
- the third digit is the sum of the first and second digits,
- and the last digit is three times the second digit?

Find the two digit positive integer that is equal to:

- one more than eight times the sum of its digits

AND

- two more than the product of 13 and the positive difference between its digits.

\[ \large{\begin{array}{ccccccc} && & & \color{blue}X& \color{blue}X & \color{blue}X&\color{blue}X\\ && & & \color{green}Y& \color{green}Y & \color{green}Y&\color{green}Y\\ +&& & & \color{red}Z& \color{red}Z & \color{red}Z&\color{red}Z\\ \hline & & & \color{green}Y& \color{blue}X& \color{blue}X & \color{blue}X&\color{red}Z\\ \hline \end{array}} \]

If \(\color{blue}X\), \( \color{green}Y\) and \(\color{red}Z\) are distinct digits in the sum above, then find \(\color{red}Z\).

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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