Download Euclidean Algebra - Rosshall Academy

Euclidean Algebra
Can you find integers x and y which satisfy the equation
2695x + 1260y = 35
First use the Euclidean algorithm to find the greatest common divisor
2695 = 2 * 1260 + 175
1260 = 7 * 175 + 35
175 = 5 * 35 + 0
The gcd (2695, 1260) = 35
Now taking the algorithm it is possible to work backwards. Start at the second last step of
the algorithm and work back each step eliminating the remainders by substituting.
Working down
Working up
35 = -7 * 2695 + 15 * 1260
2695 = 2 * 1260 + 175
35 = 1260 – 7(2695 – 2 * 1260)
1260 = 7 * 175 + 35
35 = 1260 – 7 * 175
175 = 5 * 35 + 0
So 35 = -7 * 2695 + 15 * 1260 and x = -7 and y = 15
Try these…
1. Calculate the greatest common divisor of 1666 and 418 and express it in the form
1666x + 418y
2. Show that 683 and 418 are relatively prime (greatest common divisor = 1)
Use this fact to express 1 as the sum of multiples of 763 and 662
3. 5688x + 244y = 4. Assuming x and y are integers, find their values.