Number system


Let t be the total number of CD's that Leela and Madan together sold. Then they obtain t2 rupees together.
Since Leela is the first one to take 10 rupees and also the last one to take 10 rupees, we must have t2 = 10(an old number) + (a number less than 10).
Suppose t = 10q + r, where r is the remainder when t is divided by 10. Then t2 = 100q2 + 20qr + r2. Comparing, we conclude that
r2 = 10(an odd number) + (a number less than 10).
But we know that 0 ≤ r < 10. Taking r = 0, 1, 2,.....9, we see that r = 4 or 6 (for other value of r, tens place in r2 is even). But then r2 = 16 or 36. Hence the amount left for Madan at the end is 6 rupees.
[ABC] = BC2 = [OBC] , since ODE and OCB are also simil