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Diophantine Equations

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

Diophantine Equations

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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?