Two numbers have product 120 and the sum of their squares is 289. Find the sum of the two numbers.

Introduction / Context:
Instead of solving for each number individually, this problem invites you to use algebraic identities that connect sums, products, and sums of squares. Recognizing and applying these identities leads to a direct solution for the required sum.

Given Data / Assumptions:

• Let the numbers be a and b.
• ab = 120.
• a^2 + b^2 = 289.
• We need S = a + b.

Concept / Approach:
Use the identity (a + b)^2 = a^2 + b^2 + 2ab. This provides a direct route to the square of the sum without finding a or b individually. Take the positive square root for the sum since a + b is nonnegative in typical aptitude contexts (and options reflect a positive result).

Step-by-Step Solution:

Verification / Alternative check:
If desired, solve the quadratic t^2 − 23t + 120 = 0 to find potential pairs (t represents either number). The roots multiply to 120 and sum to 23, confirming the result.

Why Other Options Are Wrong:
20 and 22 do not satisfy the identity with the given product and sum of squares. 169 is not a plausible sum here; it appears to confuse the sum of squares with the sum itself.

Common Pitfalls:
Forgetting the 2ab term in the identity or taking the wrong square root without checking feasibility.

Final Answer:
23