The sum of all three-digit numbers which give 2 as the remainder when divided by 3, is equal to:

Question

Accepted Answer

To determine the sum of all three-digit numbers that give a remainder of 2 when divided by 3, we need to identify the sequence of numbers that meet this criterion and then calculate their sum.

### Step-by-Step Explanation:

1. **Identify the Sequence:**
   - The smallest three-digit number is 100. When 100 is divided by 3, the remainder is 1. Therefore, the smallest three-digit number that gives a remainder of 2 when divided by 3 is 101.
   - The largest three-digit number is 999. When 999 is divided by 3, the remainder is 0. Therefore, the largest three-digit number that gives a remainder of 2 when divided by 3 is 998.

2. **Formulate the Sequence:**
   - The sequence of numbers that give a remainder of 2 when divided by 3 is an arithmetic sequence starting from 101 and ending at 998, with a common difference of 3 (i.e., 101, 104, 107, ..., 998).

3. **Determine the Number of Terms:**
   - The general formula for the nth term of an arithmetic sequence is given by:
     $$
     a_n = a_1 + (n-1) \cdot d
     $$
     where $a_1$ is the first term, $d$ is the common difference, and $a_n$ is the nth term.
   - Setting $a_n = 998$, $a_1 = 101$, and $d = 3$, we solve for $n$:
     $$
     998 = 101 + (n-1) \cdot 3
     $$
     $$
     998 - 101 = (n-1) \cdot 3
     $$
     $$
     897 = (n-1) \cdot 3
     $$
     $$
     n-1 = 299
     $$
     $$
     n = 300
     $$
   - Therefore, there are 300 terms in this sequence.

4. **Calculate the Sum:**
   - The sum $S_n$ of an arithmetic sequence is given by:
     $$
     S_n = \frac{n}{2} \cdot (a_1 + a_n)
     $$
   - Substituting the known values:
     $$
     S_{300} = \frac{300}{2} \cdot (101 + 998)
     $$
     $$
     S_{300} = 150 \cdot 1099
     $$
     $$
     S_{300} = 164850
     $$

### Conclusion:
- The sum of all three-digit numbers that give a remainder of 2 when divided by 3 is 164850, which corresponds to Option 1.

### Explanation of Incorrect Options:
- **Option 2 (164853):** This value is slightly higher than the correct sum. It could result from a miscalculation or an incorrect number of terms.
- **Option 3 (164847):** This value is slightly lower than the correct sum. It might result from an error in calculating the sum or using incorrect endpoints.
- **Option 4 (164844):** This value is also lower than the correct sum, likely due to similar reasons as Option 3.

Thus, Option 1 is the correct answer, as it accurately represents the sum of the specified sequence of numbers.