Two digits have a sum of 15 and their difference is 3 (order unspecified). What is the product of the two digits?

Difficulty: Easy

54
56
63
42
Cannot be determined

Correct Answer: 54

Explanation:

Introduction:
This is a simple linear system about two digits. We know the sum and the absolute difference, which uniquely determines the pair (up to order). Once the digits are found, compute their product.

Given Data / Assumptions:

Let the digits be a and b with 0 ≤ a,b ≤ 9.
a + b = 15.
|a − b| = 3.

Concept / Approach:
Solve the system using simultaneous equations. Consider both possibilities for the difference: a − b = 3 or b − a = 3. Either way, the product is the same.

Step-by-Step Solution:

Case 1: a − b = 3. Then adding with a + b = 15 gives 2a = 18 → a = 9, b = 6 → product = 54.Case 2: b − a = 3. Then adding with a + b = 15 gives 2b = 18 → b = 9, a = 6 → product = 54.Thus, regardless of order, product is 54.

Verification / Alternative check:
Try both pairs (9, 6) and (6, 9). Sum 15, difference 3, product 54 in both cases.

Why Other Options Are Wrong:
56, 63, 42 do not correspond to the product of any pair of digits whose sum is 15 and difference is 3; “Cannot be determined” is incorrect because the product is fixed.

Common Pitfalls:
Thinking order affects product (it does not) or miscalculating the difference constraint.

Two digits have a sum of 15 and their difference is 3 (order unspecified). What is the product of the two digits?

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