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.
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.
- [KVPY 2010]
(d)
Let the total number of CD's sold by the Leela and Madan together $=x$ Total money obtained by them
$=(x \times x)=x^2$
They divided $x^2$ in such that, $x^2=10$ (an odd number) $+$ a number less than $10$
$\Rightarrow \quad x=10 q+r \quad[\because 0 \leq r < 10]$
$\Rightarrow \quad x^2=(10 q+r)^2$
$\Rightarrow \quad x^2=100 q^2+20 q r+r^2$
$r^2=10$ (an odd number) $+$ a number less
than $10$
$r=16$ or $36$
$r^2=10+6$ or $3(10)+6$
Hence, the amount left for Madan at the end is $6$ rupees.
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