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The problem states:

- Bumrah and Siraj alternate to complete a piece of work in **40 days**. Bumrah takes **p** days to finish the work alone, and Siraj takes **q** days. We are asked to find how many ordered pairs of integers \((p, q)\) are possible.

### Let's break down the problem:

- Bumrah works for a day, then Siraj works for a day, and so on. 
- The total work done by both in 1 day is:
  \[
  \frac{1}{p} + \frac{1}{q}
  \]
  This represents the part of work they complete together in 1 day when working alternately.

- Since they alternate, the combined work of **2 days** will be:
  \[
  \frac{1}{p} + \frac{1}{q}
  \]
  Therefore, in 40 days, they alternate for 20 cycles (each cycle is 2 days).

- In 40 days, the fraction of work completed by them will be:
  \[
  20 \times \left(\frac{1}{p} + \frac{1}{q}\right) = 1 \quad \text{(since they complete the entire work in 40 days)}
  \]

- Simplifying, we get:
  \[
  \frac{20}{p} + \frac{20}{q} = 1
  \]

- Multiplying through by \(pq\) to eliminate the fractions:
  \[
  20q + 20p = pq
  \]
  Rearranging this equation:
  \[
  pq - 20p - 20q = 0
  \]
  Adding 400 to both sides to factorize:
  \[
  pq - 20p - 20q + 400 = 400
  \]
  This gives:
  \[
  (p - 20)(q - 20) = 400
  \]

Now, we need to find how many ordered pairs of integers \((p, q)\) are possible by solving \((p - 20)(q - 20) = 400\). 

The number of divisors of 400 will give us the number of factorizations of the equation above.

### Divisors of 400:
The prime factorization of 400 is:
\[
400 = 2^4 \times 5^2
\]
The number of divisors is given by:
\[
(4 + 1)(2 + 1) = 5 \times 3 = 15
\]
So, there are 15 divisors of 400.

### Conclusion:
Since each divisor gives us a pair \((p, q)\), there are **15 ordered pairs** of integers \((p, q)\).

Thus, the correct answer is (D) **15**.

---

Do you want further details or have any questions about this problem? 

Here are 5 related questions to extend your understanding:
1. How do you find the prime factorization of a number?
2. What is the formula for finding the number of divisors of a number based on its prime factorization?
3. How would the answer change if Bumrah and Siraj alternated working every half-day instead of every day?
4. How can you use algebraic manipulation to solve similar work-time problems?
5. What role does symmetry play in finding factor pairs of numbers?

**Tip:** Always try to express work-rate problems in terms of "work done per unit time" to simplify calculations.

Two workers Bumrah and Siraj can complete a piece of work in 40 days if they work alternately starting with Bumrah and followed by Siraj. Bumrah takes ‘p’ days to complete the same work working alone while Siraj takes ‘q’ days to complete the same work working alone. If p and q are integers, then how many ordered pairs of (p, q) are possible?

Solve a work rate problem where two workers, Bumrah and Siraj, alternate work to complete a task in 40 days. Using algebraic methods and factorization, find how many ordered pairs of integers (p, q) are possible. Includes detailed explanations and step-by-step solution for high school students.

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