Leela and Madan pooled their music CD's and sold them. They got as many rupees for each CD as the total number of CD's they sold. They share the money as follows Leela first takes 10 rupees, then Madan takes 10 rupees and they continue taking 10 rupees alternately till Madan is left out with less than 10 rupees to take. Find the amount that is left out for Madan at the end, with justification.

Answer

0

Explanation

Let t be the total number of CD's that Leela and Madan together sold. Then they obtain `t^(2)` rupees together. Since Leela is the first one to take 10 rupees and also the last one to take 10 rupees, we must have
`t^(2)=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 `t^(2)=100q^(2)+20qr+r^(2)`.
Comparing, we conclude that
`r^(2)`=10(an odd number)+(a number less than 10).
But we know that `le r le10`. Taking r=0, 1, 2,.....9, we see that r=4 or 6 (for other value of r, tens place in `r^(2)` is even). But then `r^(2) = 16` or 36. Hence the amount left for Madan at the end is 6 rupees.