IIT JEE , AIEEE Forum - Mathematics , Algebra

elastiboysai

When m=n

its a trivial case a square

so no. of squares it passes through is m

When m not equal to n

Now consider ur rectangle at the origin

The equation of diagonal is y=(m/n)*x

Now position the rectangle such that m>n

All the lines are of the form y= constant or x= constant

and the different lines have areas bounded between them

All you need now is

solutions of the equation y=(m/n)*x

where either of the coordinate is an integer

This can be generalised to y=m+n-1 when m and n are coprime

[1 for (m,n)]

this follows because if coprime

u cant find another ratio p/q = m/n

such that p and q are integers

when m not equal to n and they have a common factor

write 1,2,3,4,5,...m and 1,2,3,4,5...n

find all those points that satisfy y=(m/n)*x where both are integers

(called lattice points)

let the no. of such points be p

now answer in case 2 is m+n-1-p

for a good example

consider a rectangle sides 6/ 3

y= 6/3*x is the equation of the diagonal

now lattice points are (1,2) (2,4) =2

so no of squares the diagonal passes thru is

6+3-1-2 =6

when both are coprime

consider y=(5/3)*x

accordin 2 the logic no of squares= 5+3-1 =7