Number System — Find the number x that satisfies (x/21) × (x/189) = 1. Choose the value that makes the product exactly one.

Introduction / Context:
This problem is a straightforward algebraic equation in multiplicative form. Recognizing that the product equals 1 allows us to combine factors into x^2 and solve using square roots. Integer factorization helps confirm the perfect-square nature of the result.

Given Data / Assumptions:

• (x/21) × (x/189) = 1.
• x is real; we seek the positive value consistent with the context.
• 21 and 189 are positive integers with simple factorization.

Concept / Approach:
Multiply the numerators and denominators: (x × x) / (21 × 189) = 1 → x^2 = 21 × 189. Factor the right-hand side to test if it is a perfect square and then take the square root to find x.

Step-by-Step Solution:
1) Combine: x^2 / (21 × 189) = 1 → x^2 = 21 × 189.2) Compute 21 × 189 = 21 × (200 − 11) = 4200 − 231 = 3969.3) Note 3969 = 63^2 (since 60^2 = 3600 and 63^2 = 3969).4) Therefore, x = 63 (taking the positive root consistent with the setting).

Verification / Alternative check:
Substitute x = 63: (63/21) × (63/189) = 3 × 1/3 = 1. The identity holds exactly.

Why Other Options Are Wrong:
21 gives product 1/3; 147 gives 7/3; 3969 or 84 produce values far from 1. Only 63 satisfies the equation.

Common Pitfalls:
Not combining the fractions first; taking x^2 = 210 or 189 by mistake; forgetting to verify by substitution.

Final Answer:
63