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

In case I we have x square in horizontal line and y squares in vertical line so n = xy

(total number of square)

Similarly n + 76 = uv

The ratio of horizontal and vertical number of square in both cases will be same.

Denote HCF (x, u) = a

HCF (y, v) = b

then we can write x = ac

u = ad where HCF (c, d) = 1

Similarly y = bc

v = bd where HCF (c, d) = 1

So, uv – xy = (d² – c²) ab = 76

(d + c) (d – c) ab = 76 = 2² × 19

=> d – c = 1, d + c = 19

ab = 4

d = 10

c = 9, ab = 4

Hence n = xy = c²ab

= 9² × 4 = 324.