Find the sum of digits of the least positive number that leaves remainders 33 when divided by 52, 59 when divided by 78, and 98 when divided by 117.

Introduction:
This follows the same remainder alignment idea. When each remainder equals modulus minus the same offset, the least solution is the common period minus that offset. We then sum the digits of the resulting number.

Given Data / Assumptions:

• N mod 52 = 33
• N mod 78 = 59
• N mod 117 = 98

Concept / Approach:
Note that 33 = 52 - 19, 59 = 78 - 19, 98 = 117 - 19, so N is congruent to -19 modulo each modulus. Compute LCM and then subtract 19 to get the least positive N.

Step-by-Step Solution:

Verification / Alternative check:
Check: 449 mod 52 = 33, 449 mod 78 = 59, 449 mod 117 = 98. All remainders match.

Why Other Options Are Wrong:
19, 21, 27, and 36 are not the digit sum of the least valid N, which equals 449.

Common Pitfalls:
Missing the shared offset or taking a non-minimal solution by adding an extra multiple of the LCM.

Final Answer:
17