If *P*, *Q*, and *R* are natural numbers and satisfy the equation:

$$PQ = R + Q$$

Which of the following statements or combinations of statements are sufficient to find the values of *P*, *Q*, and *R*?

Statements:

(A) $$P + R = 8 + Q$$

(B) $$Q^2 = \frac{R^2}{P+1}$$

(C) $$Q = P + 2$$

Question

If *P*, *Q*, and *R* are natural numbers and satisfy the equation:

$$PQ = R + Q$$

Which of the following statements or combinations of statements are sufficient to find the values of *P*, *Q*, and *R*?

Statements:

(A) $$P + R = 8 + Q$$

(B) $$Q^2 = \frac{R^2}{P+1}$$

(C) $$Q = P + 2$$

Accepted Answer

**Solution:**

1. Given the equation:
   $$PQ = R + Q$$

2. From statement (A):
   $$P + R = 8 + Q \ R = 8 + Q - P$$

Substituting into the main equation:
   $$PQ = (8 + Q - P) + Q = 8 + 2Q - P$$

This alone is not sufficient to find unique values of *P*, *Q*, and *R*.

3. From statement (B):
   $$Q^2 = \frac{R^2}{P+1} \ Q = \frac{R}{\sqrt{P+1}}$$

Using the main equation:
   $$P 	imes Q = R + Q \ P 	imes \frac{R}{\sqrt{P+1}} = R + \frac{R}{\sqrt{P+1}}$$

Simplifying this leads to a value for *P*:
   $$P = 3$$

Substituting back:
   $$Q = \frac{R}{2}$$

Using the main equation:
   $$3 	imes \frac{R}{2} = R + \frac{R}{2} \ \frac{3R}{2} = \frac{3R}{2}$$

This is an identity and does not determine unique values.

4. From statement (C):
   $$Q = P + 2$$

Alone, this is insufficient.

5. Combining (B) and (C):
   Substitute $Q = P + 2$ into $Q = \frac{R}{\sqrt{P+1}}$:
   $$P + 2 = \frac{R}{\sqrt{P+1}}$$

Using the main equation and these two, we can solve for *P*, *Q*, and *R* uniquely.

6. Similarly, combining (A) and (C) or (A) and (B) provides three equations with three variables, allowing unique solutions.

**Answer:** Either A and B or A and C are sufficient to answer the question.

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