Number linear combinations

2026/5/1

Enter two numbers (preferably non-zero positive integers) in this project to get started. Press space if you ever want to restart with two different numbers entered.

The GCD (Greatest Common Divisor) of two numbers is the biggest number that both numbers are a multiple of. This can be found using a method called the Euclidean algorithm, which repeatedly replaces the bigger number with its value mod the smaller one. This continues until one of the numbers becomes 0, where the other one will be the GCD of the original two numbers.

In this project's equations, dividends are orange, divisors are yellow, quotients are blue and remainders are purple. The divisor and remainder of each equation are used as the dividend and divisor of the next, unless the remainder that would've become a divisor is 0. You can't divide by 0.

Press the left and right arrow keys to view alternate equations. Page 2 has each equation rewritten to have the remainder as the subject. Page 3 has each remainder rewritten as multiples of the original two numbers added together. These multipliers come from the previous remainder's page 2 equation and are modified based on that equation's quotient, dividend's multipliers and divisor's multipliers.

At the bottom of the screen is a geometric visualisation of the page 1 equations used for finding the GCD, which is the biggest edge length for a square that can perfectly tile a rectangle whose dimensions are two numbers. The biggest possible square is tested and if there's leftover space, the GCD will be equal to the GCD of that rectangular gap and the Euclidean algorithm is continued recursively.