Given (PQ) * (RQ) = XXX where each letter denotes a distinct digit and (PQ), (RQ) are two–digit numbers sharing the same units digit Q. Determine P + Q + R + X. I. X = 9. II. All digits represented by letters are unique.

Introduction / Context:
We are told two two-digit numbers, PQ and RQ (same units digit Q), multiply to a three-digit number whose digits are all equal to X (i.e., XXX). The task is not to compute values outright but to decide which statements are sufficient to determine P + Q + R + X uniquely in a data-sufficiency sense.

Given Data / Assumptions:

• P, Q, R, X are digits 0–9; (PQ) and (RQ) are valid two-digit numbers, so P and R are non-zero.
• Statement I: X = 9, hence product equals 999.
• Statement II: All four letters represent distinct digits.

Concept / Approach:
Use factorization of 999 to test if there exist two two-digit factors sharing the same units digit. If a unique pair emerges, we can read off P, Q, R and sum with X.

Step-by-Step Solution:
From I: XXX = 999. Factor 999 = 27 * 37 (other factor pairs are 1*999, 3*333, 9*111 which are not both two-digit).The only two two-digit factor pair is 27 and 37, which share the units digit 7. Thus Q = 7, P = 2, R = 3, X = 9.Therefore P + Q + R + X = 2 + 7 + 3 + 9 = 21, uniquely determined by Statement I alone.Statement II alone gives only “digits are unique,” which does not pin down the number; many possibilities remain.

Verification / Alternative check:
No other two-digit factorization of 999 exists; hence uniqueness is guaranteed without needing Statement II.

Why Other Options Are Wrong:
II alone is insufficient; “both together” is unnecessary because I already suffices; “either alone” is false since II alone fails.

Common Pitfalls:
Overlooking that 999 has exactly one two-digit factor pair and that both factors end with 7, satisfying the shared-units requirement.

Final Answer:
Statement I alone is sufficient.