A pen costs Rs 11 and a notebook costs Rs 13. Find the number of ways in which a person can spend exactly Rs 1000 - Solution 2

You can check out a simpler solution here.
This one is a bit more complicated and uses modulo arithmetic.

The problem states:

• A pen costs ₹11.
• A notebook costs ₹13.
• We need to find the number of ways to spend exactly ₹1000 on these items.

Step 1: Formulating the equation

Let:

• $x$ be the number of pens.
• $y$ be the number of notebooks.

Then the equation to satisfy is:

$$11x + 13y = 1000$$

where $x, y$ are non-negative integers.

Step 2: Finding integer solutions

We solve for $x$ in terms of $y$:

$$x = \frac{1000 - 13y}{11}$$

For $x$ to be an integer, $(1000 - 13y)$ must be divisible by 11.

Let's determine valid values for $y$ by checking divisibility:

$$1000 - 13y \equiv 0 \pmod{11}$$

Since $1000 \equiv 10 \pmod{11}$ and $13y \equiv 2y \pmod{11}$, we get:

$$10 - 2y \equiv 0 \pmod{11}$$ $$2y \equiv 10 \pmod{11}$$

Multiplying both sides by the modular inverse of 2 modulo 11 (which is 6):

$$y \equiv 10 \times 6 \pmod{11}$$ $$y \equiv 60 \equiv 5 \pmod{11}$$

Thus, $y = 5, 16, 27, \dots$ as long as $13y \leq 1000$.

Step 3: Finding valid values of $y$

The maximum $y$ should satisfy:

$$13y \leq 1000 \Rightarrow y \leq \frac{1000}{13} \approx 76.92$$

So, the possible values of $y$ are:

$$y = 5, 16, 27, 38, 49, 60, 71$$

Counting these, we get 7 valid values.

Conclusion:

Thus, the number of ways to spend exactly ₹1000 on pens and notebooks is 7.