Find the sum of digits of the least positive number that leaves remainders 35 when divided by 54, 61 when divided by 80, and 100 when divided by 119.

Introduction:
This is a simultaneous congruence problem. When all remainders align as the same negative offset from each modulus, the least positive solution is the least common multiple of the moduli minus that common offset. We then sum the digits of the resulting least number.

Given Data / Assumptions:

• N mod 54 = 35
• N mod 80 = 61
• N mod 119 = 100

Concept / Approach:
Recognize that 35 = 54 - 19, 61 = 80 - 19, 100 = 119 - 19, so N is congruent to -19 modulo each modulus. Therefore N = LCM(54, 80, 119) - 19 gives the least positive solution.

Step-by-Step Solution:

Verification / Alternative check:
Check: 257021 mod 54 = 35, mod 80 = 61, mod 119 = 100, confirming correctness.

Why Other Options Are Wrong:
19, 23, 26, and 28 are plausible digit sums but do not match the sum of digits of the true least solution 257021.

Common Pitfalls:
Solving each congruence separately without noticing the common negative offset, which significantly simplifies the Chinese remainder reasoning.

Final Answer:
17