Which of the following numbers is divisible by 24?

Introduction / Context:
This question tests divisibility rules, specifically for determining quickly whether a number is divisible by 24. Since 24 factors as 8 × 3, a number is divisible by 24 exactly when it is divisible by both 8 and 3. Using divisibility tests for 8 and 3 allows us to check each option without performing full long division, which saves time in competitive exams.

Given Data / Assumptions:
- We need to find which one of the numbers 37804, 2015736, 36718, and 53810 is divisible by 24.
- 24 = 8 × 3, so divisibility by 24 requires divisibility by 8 and by 3 simultaneously.
- We will apply standard divisibility tests: for 8, check the last three digits; for 3, check the sum of digits.

Concept / Approach:
A number is divisible by 8 if its last three digits form a number divisible by 8. A number is divisible by 3 if the sum of its digits is a multiple of 3. We apply these tests to each candidate. Only numbers that pass both tests are divisible by 24. We then match the correct candidate with the answer choices.

Step-by-Step Solution:
Step 1: Check 37804. Last three digits are 804. 804 ÷ 8 = 100.5, so it is not divisible by 8. Therefore 37804 is not divisible by 24. Step 2: Check 2015736. Last three digits are 736. 736 ÷ 8 = 92 exactly, so it is divisible by 8. Now sum its digits: 2 + 0 + 1 + 5 + 7 + 3 + 6 = 24. Since 24 is divisible by 3, 2015736 is divisible by 3. Hence 2015736 is divisible by both 8 and 3, so it is divisible by 24. Step 3: Check 36718. Last three digits are 718. 718 ÷ 8 = 89.75, so 36718 is not divisible by 8 and therefore not by 24. Step 4: Check 53810. Last three digits are 810. 810 ÷ 8 = 101.25, so not divisible by 8. Hence 53810 is not divisible by 24. Step 5: Only 2015736 passes both divisibility tests.

Verification / Alternative check:
We can confirm by direct division for 2015736: dividing by 24 stepwise or using a calculator gives an integer quotient with no remainder. Since the other numbers already fail one of the necessary tests (for 8), they cannot be divisible by 24. Therefore 2015736 is the only correct choice.

Why Other Options Are Wrong:
Options 37804, 36718, 53810: Each of these fails the divisibility test for 8. Even if some might be divisible by 3, failing divisibility by 8 means they are not divisible by 24.
Option None of these: Incorrect because 2015736 is in fact divisible by 24 and is one of the listed options.

Common Pitfalls:
A typical error is checking divisibility by 3 alone and ignoring divisibility by 8, or vice versa. Another mistake is checking all digits for divisibility by 8 instead of just the last three. Remembering that 24 factors into 8 and 3, and applying both tests properly, is the key to solving such problems quickly and correctly.

Final Answer:
The number 2015736 is divisible by 24.