Question

Given below is an equation where the letters represent digits: \[ (PQ)\,(RQ) = XXX \] Determine the sum of $P+Q+R+X$. % Statement I $X = 9$. % Statement II The digits are unique.

• A. Statement I alone is sufficient to answer the question.
• B. Statement II alone is sufficient to answer the question.
• C. Statement I and Statement II together are sufficient, but neither alone is sufficient.
• D. Either Statement I or Statement II alone is sufficient.
• E. Neither Statement I nor Statement II is necessary to answer the question.

Hint:

For digit-repetition problems ($111, 222, \dots, 999$), always test divisibility by 37. Repdigits in base 10 follow the pattern $111 = 3 \times 37$, $222 = 6 \times 37$, and so on. This trick quickly reduces the search space.

Answer 1

Answer: (E)

Step 1: Understand the structure.
We are told: \[ (PQ)(RQ) = XXX \] where $PQ$ and $RQ$ are two-digit numbers, and $XXX$ is a three-digit number with all digits equal. So $XXX$ can be $111, 222, 333, \dots, 999$.

Step 2: Recognize the multiples of 37 pattern.
Numbers with all digits equal (repdigits) are always divisible by 37: \[ 111 = 3 \times 37, \quad 222 = 6 \times 37, \quad 333 = 9 \times 37, \dots, 999 = 27 \times 37 \] Thus, the equation becomes: \[ (PQ)(RQ) = k \times 37 \quad \text{for some integer } k \]

Step 3: Possible values.
Check each repdigit: - $111 = 3 \times 37$ - $222 = 6 \times 37$ - $333 = 9 \times 37$ - $444 = 12 \times 37$ - $555 = 15 \times 37$ - $666 = 18 \times 37$ - $777 = 21 \times 37$ - $888 = 24 \times 37$ - $999 = 27 \times 37$ So each repdigit $XXX$ corresponds to some product.

Step 4: Unique consistency.
Among these nine cases, we must check which can be represented as $(PQ)(RQ)$ where $PQ$ and $RQ$ are valid two-digit numbers. It turns out that only for $999$ the unit’s digit condition matches consistently (because both sides end in the same digit). Thus: \[ XXX = 999 \]

Step 5: Digits identified.
So, $X=9$. This gives: \[ (PQ)(RQ) = 999 \] Now $P, Q, R$ can be uniquely determined (not shown in detail here as uniqueness is guaranteed).

Step 6: Data sufficiency check.
- Statement I says $X=9$, but we already derived $X=9$ from the structure itself. So it is not needed. - Statement II says digits are unique, but this too is not needed — uniqueness follows automatically from the multiplication structure. Therefore, the problem can be solved

without either statement.

Final Answer: \[ \boxed{\text{E) Neither Statement I nor Statement II is necessary to answer the question.}} \]