Beelines

Very few people seem to have attempted this problem, so here is a little help:

?	examples of coprime positive integers:
5 and 12
16 and 15
3 and 10
	they have no common factors.

?	examples of positive integers that are not coprime:
25 and15 (5 is a common factor)
12 and 16 (2 and 4 are common factors)
3 and 12 (3 is a common factor)

?	starting at (0,0)
a straight line to (2,1) crosses 2 squares
a straight line to (3,1) crosses 3 squares
a straight line to (x,1) crosses ? squares
a straight line to (p,q) crosses ? squares
						Why?