The sum of the digits of a two-digit number is 15 and the difference between the digits is 3. What is the two-digit number?

### Concept & Logic Linear equations with two variables. The difference between digits implies two absolute scenarios depending on which digit is larger. ### Step-by-Step Solution Let the two digits be $x$ and $y$. Condition 1: Sum of digits is 15. $x + y = 15$ Condition 2: Difference between digits is 3. This gives two possibilities: Case A: $x - y = 3$ Case B: $y - x = 3$ Solving Case A: Add $x + y = 15$ and $x - y = 3$. $2x = 18 \Rightarrow x = 9$ $y = 15 - 9 = 6$ If $x$ is the ten's digit, the number is 96. Solving Case B: Add $x + y = 15$ and $y - x = 3$. $2y = 18 \Rightarrow y = 9$ $x = 15 - 9 = 6$ If $x$ is the ten's digit, the number is 69. Since the question does not specify which digit is larger (ten's or unit's), both 69 and 96 are valid possibilities. Thus, a unique number cannot be found. ### Exam Strategy & Shortcut Look at the options immediately. Options (a) 69 and (c) 96 both satisfy the conditions ($6+9=15$ and $|6-9|=3$). Since both are present and valid, the unique answer must be "Cannot be determined". ### Common Pitfall Assuming that the ten's digit must be larger than the unit's digit (or vice versa) and prematurely selecting either 69 or 96 without checking both cases. ### Final Answer Therefore, the correct answer is **Cannot be determined**.