Which of the following numbers is divisible by 3?

Concept: The divisibility rule for 3 states that a number is exactly divisible by 3 if and only if the sum of its digits is divisible by 3. This holds because every power of 10 leaves remainder 1 when divided by 3, so a number's remainder on division by 3 equals its digit sum's remainder on division by 3.

Application: Applying this rule to each option:

1. 2345: digit sum = 2 + 3 + 4 + 5 = 14. Since 14 is not divisible by 3, 2345 is not divisible by 3.

2. 9875: digit sum = 9 + 8 + 7 + 5 = 29. Since 29 is not divisible by 3, 9875 is not divisible by 3.

3. 6504: digit sum = 6 + 5 + 0 + 4 = 15. Since 15 is divisible by 3 (15 = 3 x 5), 6504 is divisible by 3.

4. 9833: digit sum = 9 + 8 + 3 + 3 = 23. Since 23 is not divisible by 3, 9833 is not divisible by 3.

Cross-check: Direct division confirms this: 6504 divided by 3 equals 2168 exactly, an integer, independently verifying the digit-sum result.

Result: Only 6504 satisfies the divisibility condition, so 6504 is the number divisible by 3 among the given options.