Linear Diophantine Equations: Level 2 Challenges Practice Problems Online

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.

$\large{\begin{array}{ccccccc} && & & \color{#3D99F6}X& \color{#3D99F6}X & \color{#3D99F6}X&\color{#3D99F6}X\\ && & & \color{#20A900}Y& \color{#20A900}Y & \color{#20A900}Y&\color{#20A900}Y\\ +&& & & \color{#D61F06}Z& \color{#D61F06}Z & \color{#D61F06}Z&\color{#D61F06}Z\\ \hline & & & \color{#20A900}Y& \color{#3D99F6}X& \color{#3D99F6}X & \color{#3D99F6}X&\color{#D61F06}Z\\ \hline \end{array}}$

If $\color{#3D99F6}X$ , $\color{#20A900}Y$ and $\color{#D61F06}Z$ are distinct digits in the sum above, then find $\color{#D61F06}Z$ .

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!

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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Linear Diophantine Equations: Level 2 Challenges