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Olympiad workbook
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Olympiad workbook
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IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

A pen costs ₹ 11 and a notebook costs ₹ 13 . Find the number of...

A pen costs ₹11₹ 11 and a notebook costs ₹13₹ 13. Find the number of ways in which a person can spend exactly ₹ 1000 to buy pens and notebooks.

Correct answer

7

Step-by-step solution

\cos t of a pen is ₹11₹ 11 \cos t of a notebook is ₹ 13 11x+13y=1000y=1000−11x13∈I12−11x13∈I⇒(13−1)−(13x−2x)13∈I2x−113∈I⇒12x−613∈I13x−(x+6)13∈I⇒x+613∈Ix=13λ−6(1≤x≤90)x∈{7,20,……,85}{λ={1,2,…,3}}\begin{aligned} & 11 x+13 y=1000 \\ & y=\frac{1000-11 x}{13} \in I \\ & \frac{12-11 x}{13} \in I \Rightarrow \frac{(13-1)-(13 x-2 x)}{13} \in I \\ & \frac{2 x-1}{13} \in I \Rightarrow \frac{12 x-6}{13} \in I \\ & \frac{13 x-(x+6)}{13} \in I \Rightarrow \frac{x+6}{13} \in I \\ & x=13 \lambda-6 \quad(1 \leq x \leq 90) \\ & x \in\{7,20, \ldots \ldots, 85\} \quad\{\lambda=\{1,2, \ldots, 3\}\} \end{aligned}
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