Find all numbers between 300 and 400 that leave a remainder of 4 when divided by 12, 18, and 36. Then compute the sum of those numbers.

Introduction / Context:
We are told of a number N between 300 and 400 that leaves remainder 4 upon division by 12, 18, and 36. Because 36 is a multiple of both 12 and 18, the single congruence modulo 36 controls the condition. The task then asks for the sum of all such numbers in the interval (there are two), not just one value.

Given Data / Assumptions:

• N is strictly between 300 and 400.
• N ≡ 4 (mod 12), N ≡ 4 (mod 18), and N ≡ 4 (mod 36).
• LCM(12, 18, 36) = 36, so it suffices that N ≡ 4 (mod 36).

Concept / Approach:
All solutions are of the form N = 4 + 36k. Choose integers k such that 300 < N < 400. List the valid N and then add them. Ensure 400 is excluded if the wording requires strictly between 300 and 400.

Step-by-Step Solution:

Verification / Alternative check:
Check remainders: 328 mod 36 = 4; 364 mod 36 = 4. Both also leave remainder 4 modulo 12 and 18 since 36 encompasses those moduli. 400 is excluded as it equals the upper boundary; while 400 mod 36 = 4, it is not strictly between 300 and 400.

Why Other Options Are Wrong:

• 840, 324, 1024, 728: Do not match the computed sum of valid values in the specified interval.

Common Pitfalls:

• Including 400 in the sum despite the phrase “between 300 and 400.”
• Using LCM(12, 18) instead of LCM(12, 18, 36) without noticing 36 already governs both.

Final Answer:
692