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