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.