Integer equation - practice problems
An integer equation, also known as a Diophantine equation, is an equation for which only integer solutions are sought or allowed. Named after the ancient Greek mathematician Diophantus, these equations range from simple linear forms to complex polynomial expressions. Common examples include linear Diophantine equations (ax + by = c) and the famous Pythagorean equation (x² + y² = z²). Finding integer solutions often requires techniques different from standard equation solving, including modular arithmetic and number theory. Integer equations have applications in cryptography, computer science, and optimization problems with discrete variables. Some Diophantine equations, like Fermat's Last Theorem, have challenged mathematicians for centuries.Number of problems found: 194
- X and Y
X and Y are positive integers. If X+Y+XY=32, what is the difference between X and Y? - Harold
Harold made a rectangular dog run in his backyard. The area of the dog run is 96 square feet. What are three different possible dimensions of dog run? - There 35
There are three points on a straight line: A, BC. If CD = 8x, DE = 3, and CE = x + 10, what is CD? Simplify your answer and write it as a proper fraction, mixed number, or integer. - Cuboid - sum of edges length
Calculate the cuboid's dimensions if the sum of its edges is 19 cm. The body's diagonal size is 13 cm, and its volume is 144 cm³. The total surface area is 192 cm². - A group
A group of young men decides to raise ksh 480,000 to start a business. Before the actual payment was made, four members pulled out, and each of the remaining had to pay an additional ksh 20,000. Determine the original number of members. - Penny and rolls
For one hundred crowns I need to buy exactly one hundred pieces of pastry at the following prices: rolls at 0.50 crowns each, bread at 10 crowns each, and buns at 3 crowns each. How many of each should I buy? (The Slovak koruna was the currency in Slovaki - Four-digit number George is thinking of a four-digit number and gives us the following clues: a) Its digit sum equals one hundredth of the number obtained by rounding the mystery number to the nearest hundred. b) Its last digit is 1 more than the second-to-last digit. c) T
- MO Z7 2025
Adela and Susan ate plums. On the first day, Adela ate three quarters of what Susan ate that day. On the second day, Susan ate three halves of what Adela ate that day. Together, they ate 31 plums over the two days, and each girl ate a whole number of plum - Triangles - combinations
How many different triangles with sides of whole centimetres have a perimeter of 12 cm? - Fair
Several students from our class went on excursions and each paid the same amount. In the end, some money was left over that needed to be returned to them. If we returned €3 to each person, there would still be €5 left. If we returned €4 to each person, th - Red and Blue Candies
The candy store sold two types of chocolate candies. Red for 8 Sk and blue for 6 Sk. We bought 20 candies for 136 CZK. How many candies are blue and how many red? We have red candies: We have blue candies: - Plot rectangle mesh
How many different plots of land in the shape of a rectangle with length and sides in whole meters can we fence if we have 49 m of mesh available? - Spider fly legs
There are spiders and flies on the window. They have a total of 38 legs. How many spiders and flies are there if a spider has 8 legs and a fly has 6? Just give one solution. - Dolphin Shark fish
Mr. Dolphin and Mr. Shark were skilled fishermen. Once, they caught a total of 70 fish. Five-ninths of the fish Mr. Dolphin caught were trout. Two-seventeenths of the fish Mr. Shark caught were carp. How many fish did Mr. Dolphin catch? - Karel digit error Carl had to multiply two two-digit numbers. Out of care, he changed the order of the digits in one of the factors and got a product that was 4,248 less than the correct result. What is the correct result? How much should Karl have earned?
- Position of digits Find a number with six digits. If you put the last digit before the first, you get a new number that is five times larger. The digits between must not change their position.
- Three-digit - sum A three-digit number has a digit sum of 16. If we change the digits in the hundreds and tens places in this number, the number is reduced by 360. If we swap the ten's and one's digits in the original number, the number increases by 54. Find this three-dig
- Karel grade average Charles has an average grade of exactly 1.12 from five-minute episodes. Prove that at least 22 of them have one.
- Equal number multiplication When I multiply two equal natural numbers, I get the same result as when I add them together. Which ones are they?
- Smallest z9 Find the smallest positive numbers a and b for which 7a³ = 11b⁵
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