A two-digit number has its unit's digit 2 more than its ten's digit. If the product of the number and the sum of its digits is 144, what is the number?

Problem restatement
Let the number be 10x + y, with y = x + 2. Also, (10x + y) × (x + y) = 144. Find the number.

Given data

• y = x + 2
• Product with digit-sum equals 144

Concept/Approach
Express everything in x, solve the quadratic, and retrieve the digits.

Step-by-Step calculation
Number = 10x + (x + 2) = 11x + 2Digit-sum = x + (x + 2) = 2x + 2 = 2(x + 1)(11x + 2) × 2(x + 1) = 144 ⇒ (11x + 2)(x + 1) = 7211x² + 13x + 2 = 72 ⇒ 11x² + 13x − 70 = 0Discriminant = 13² + 4 × 11 × 70 = 3249 = 57²x = (−13 + 57) ÷ 22 = 2 (valid digit); so y = 4Number = 24

Verification/Alternative
Check: 24 × (2 + 4) = 24 × 6 = 144. Units digit (4) is 2 more than tens (2).

Common pitfalls
Forgetting that digits must be integers in 0–9; discarding the negative/invalid root is essential.

Final Answer
24