For a two-digit number, the sum of its digits is 15 and the difference between the two digits is 3. What is the product of its digits?
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A72
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B56
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C54
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DCannot be determined
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E63
Answer
Correct Answer: 54
Explanation
Introduction: The pair of digits is determined (up to order) by their sum and difference. Once the digits are found, compute the product; order does not affect the product.
Given Data / Assumptions:
- Let digits be a and b.
- a + b = 15.
- |a − b| = 3.
Concept / Approach: Solve the system twice (difference positive or negative); in either case, the same two digits arise in opposite order, resulting in a unique product.
Step-by-Step Solution:
Case 1: a − b = 3 → with sum 15, 2a = 18 → a = 9, b = 6 → product 54.Case 2: b − a = 3 → with sum 15, 2b = 18 → b = 9, a = 6 → product 54.Verification / Alternative check: Check both orders: {9, 6} and {6, 9}; both satisfy sum and difference, and both yield product 54.
Why Other Options Are Wrong: 72, 56, 63 are products of other pairs not meeting both constraints; “Cannot be determined” is incorrect because the product is uniquely determined.
Common Pitfalls: Assuming order affects product or miscomputing the difference condition.
Final Answer: 54