Shortcut Trick Find the LCM of (15, 24, 32, 48) which is 480. The required number format is 480k + 9. For the smallest 5-digit number (starting at 10000), 480 × 20 = 9600, so we test k ≥ 21. Check k = 21: (480 × 21) + 9 = 10080 + 9 = 10089. Validate divisibility: 10089 ÷ 19 = 531. Sum of digits: 1 + 0 + 0 + 8 + 9 = 18. ∴ The correct answer is 18. Alternate Method Given: Divisors = 15, 24, 32, 48; Common Remainder = 9; x is divisible by 19. Formula: x = LCM(Divisors) × k + Remainder ⇒ Prime factors: 15 = 3 × 5; 24 = 2^(3) × 3; 32 = 2^(5); 48 = 2^(4) × 3 ⇒ LCM = 2^(5) × 3 × 5 = 32 × 15 = 480 ⇒ x = 480k + 9 ⇒ Smallest 5-digit number ≥ 10000: 10000 ÷ 480 ≈ 20.83, so k ≥ 21 ⇒ For x to be divisible by 19: (480k + 9) ÷ 19 must leave remainder 0 ⇒ (19 × 25 + 5)k + 9 ⇒ 5k + 9 ≡ 0 (mod 19) ⇒ Testing values for k ≥ 21: If k = 21, 5(21) + 9 = 105 + 9 = 114 ⇒ 114 ÷ 19 = 6 (Exactly divisible) ⇒ x = 480 × 21 + 9 = 10080 + 9 = 10089 ⇒ Sum of digits = 1 + 0 + 0 + 8 + 9 = 18 ∴ The correct answer is 18. Related Concepts: LCM Method for Remainders When a number leaves the same remainder 'r' when divided by a, b, and c, the general form is LCM(a, b, c) × k + r. Divisibility by 19 To check if a number is divisible by 19, add twice the last digit to the remaining number. Repeat until a small number is obtained. Sum of Digits Property Used often in divisibility rules for 3 and 9; for x = 10089, the sum is 1+0+0+8+9=18, which is itself divisible by 3 and 9.