Question

Find the smallest positive integer d such that d=105m+216n, where m & n are integers.

Answer 1

Compute the GCD of 105 and 216 using thr Euclidean algorithm. A linear combination of 105 and 216 of the form

a 105 + b 216

can be reprented by the vector:

(a, b, a 105 + b 216)

We start with the two vectors:

v1 = (105, 0, 105)

v2 = (0,216, 216)

The integer part of 216/105 is 2.

v2 - 2 v1 = (-210, 216, 6)

We now define the new v2 to be the old v1 while the above vector becomes the new v1:

v1 = (-210, 216, 6)

v2 = (105, 0, 105)

We now repeat the previus step.

Integer part of 105/6 is 17.

v2 - 17 v1 = (3675, -3672, 3)

If you then would run another step, you would end up with a last component of zero, this means that the GCD is 3 and you have:

3 = 3675*105 - 3672*216

You can then add that last vector with zero last component to make the integers m and n positive.