2015/066) A line passes through A (1,1) and B (100,1000). How many other points with both coordinates integers are on this line segment between A and B.

One of the points is $A(1,1)$ and second point $B(100,1000)$ The slope is $\dfrac{999}{99}$ or in lowest form $\dfrac{111}{11}$. The equation of line is

$(y-1) = \dfrac{111}{11}(x-1)$

or $11(y-1) = 111(x-1)$

so $y-1 = 111t$ and $x-1 = 11t$

$y = 111 t + 1$ and $x = 11 t + 1$

for x and y to be integer both $111 t$ and $11t$ have to be integer and hence $t$ is integer

the 1^st point is for t = 0 and 2^nd point for t = 9

there are 8 values of t( from 1 to 8) so number of points = 8