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Diophantine Equations

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Diophantine Equations, a selection of answers from the Dr. Math archives.

Diophantine Equations
We have searched the Web for information about Diophantine equations.

Diophantine Equations, Step by Step
Find all positive integer solutions to 43x + 7y + 17z = 400.

Integer Solutions of ax + by = c
Given the equation 5y - 3x = 1, how can I find solution points where x and y are both integers? Also, how can I show that there will always be integer points (x,y) in ax + by = c if a, b and c are all integers?

Buying Cows, Pigs, and Chickens
A farmer buys 100 animals for $100. The animals include at least one cow, one pig, and one chicken. If a cow costs $10, a pig costs $3, and a chicken costs $0.50, how many of each did he buy?

How Many Mice, Cats, and Dogs?
You must spend $100 to buy 100 pets, choosing at least one of each pet. The pets and their prices are: mice @ $0.25 each, cats @ $1.00 each, and dogs @ $15.00 each. How many mice, cats, and dogs must you buy?

Money Puzzle
A man goes to the bank and asks for x dollars and y cents.

Systems with More Variables than Equations
Solve this system: 187y+98x+45z=48; 2y+9x+3z=198.

Linear Diophantine Equation
Given positive integers a and b such that a|b^2, b^2|a^3, a^3|b^4, b^4|a^5,....., prove that a = b.

Solving Multivariable Diophantine Equations
Finding general solutions to two diophantine equations.

Diophantine Equations, Apples, Bananas
Mick has $5.63 worth of bananas and apples. Bananas are 13 cents each and apples are 7 cents each. How many different combinations of apples and bananas can he have and what are they?

Diophantine equations in Number Theory
If a and b are relatively prime positive integers, prove that the Diophantine equation ax-by = c has infinitely many solutions in the positive integers.

All Possible Solutions: Diophantine Equations
Lance and Mario sold hot dogs for $1.65, hamburgers for $2.35, sodas for $.85, and a combo plate with fries, salad, a hamburger, and soda for $3.89. They sold 80 items in an hour for $163.24. How many of each kind of food did they sell, and how much did they make on each kind of food?

Second-Degree Two-Variable Diophantine Equation
Solve Ax^2+Bxy+Cy^2+Dx+Ey+F = 0 where B^2-4AC=k^2 for some integer k.

Linear Diophantine Equations
...how was the t-variable introduced, and what is the general method?

Positive Unit Fractions
Find five different positive unit fractions whose sum is 1.

Switching Dollars and Cents
How do I find an equation?

Return to
High School Linear Equations
High School Number Theory
Index of Selected Answers to Common Questions

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