3X+7Y=188 What integer solution (X,Y) yields the smallest positive difference (Y-X)?

Answer 1

x = 16, y = 20

Answer 2

3X+7Y=188 Or 7Y=188 - 3X
Or Y=188/7 - 3X/7.
So Y- X= 188/7- 3X/7 - X =188/7 - 10X/7.
This should be the smallest positive for some integer solution (X,Y)
For this to be the smallest positive, we have:
188/7 - 10X/7>0
Or 188-10X>0.
Or X < 18.8.
For (Y -X) to be to be smallest positive, Y should be the smallest and X should be maximum..
then minimum value of Y=188/7 -3X/7
=(188- 56.4)/7
=18.8
Therefore, the integer solution should be such
that X is less or equal to 18 and Y is greater than or equal to 19.
If we put X=18 in 3X+7Y=188, and solve for Y, it does not lead to an integer.
If we put X=17 in 3X+7Y=188, and solve for Y, it does not lead to an integer.
If we put X=16 in 3X+7Y=188, and solve for Y, it does lead to an integer value Y=20 which is greater than 19 as desired. Therefore, the integer solution (X,Y) with smallest positive difference is (16,20)
Answer=(16,20) and the difference Y - X = 4 .

Answer 3

This is a Diofantes equation. It's solution is of the form (I'm not telling you how to solve these, find it yourself!):

x = 7*k - 376
y = 188 - 3*k

where k is an integer. So the difference is of the form:

y - x = 564 - 10*k

In order for this difference to be positive, k must be k=<56. For k = 56, you get the smallest (the value of the expression decreases as k incerease, so it is minimal for maximum possible k) for k = 56:
y - x = 4
(x, y) = (16, 20)

Answer 4

I'm positive that the smallest difference is a solution that yields integers. I think.