Number Theory

Linear Diophantine Equations

Linear Diophantine Equations: Level 2 Challenges

         

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.

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