Question

A rectangle can be divided into 'n' equal squares. The same rectangle can also be divided into (n+76) equal squares . Find n

Answer 1

Let number of square in row is x and in column is y then, total number of square in rectangle is xy = n

similarly, in 2nd case, number of square in row is u and in column is v then, total number of square in rectangle is uv = (n + 76)

but we know, ratio of horizontal and vertical number of square in both cases will be same.
so, x/u = y/v => xv = uy

Let HCF{x, u} = a
and HCF{y, v} = b

then, we can write x = ac , u = ad
and similarly, y= bc and v = bd
where HCF {c, d} = 1

now, uv - xy = n + 76 - n = 76
(ad)(bd) - (ac)(bc) = 76
abd² - abc² = 76
ab(d² - c²) = 76
(d - c)ab(d + c) = 1 × 2² × 19
d - c = 1, ab = 4 and (d + c) = 19
d = 10, c = 9

now, n = xy = (ac)(bc) = abc² = 4 × 9²
= 4 × 36 = 144

Answer 2

→As rectangle A can be divided into 'n' equal squares and same rectangle can be divided into ''(n+76)'' squares.

Let side of square which is divided into n squares = a units

and , side of square which is divided into (n+76) squares = b units

Also,a>b

Area of square = (Side)²

→ n a²=(n+76)b²

→n a²- n b²= 76 b²

→n(a²-b²)=76 b²

→ n=$\frac{76b^{2}}{a^{2}-b^{2}}$