find the sum of all 3digits numbers ,which leave the remainder three when divided by 5.
To find the sum of all three-digit numbers that leave a remainder of three when divided by five, you can follow these steps:
1. Determine the first three-digit number that meets the given condition.
- Since we are looking for numbers that leave a remainder of three when divided by five, the first number that satisfies this condition is 103, as dividing it by 5 gives us a remainder of 3.
2. Determine the last three-digit number that meets the given condition.
- The largest three-digit number is 999. To determine the last number that satisfies the condition, we need to find the largest multiple of 5 less than or equal to 999. This can be done by dividing 999 by 5 and rounding down to the nearest whole number (quotient), then multiplying the quotient by 5. So, the largest three-digit number that leaves a remainder of three when divided by five is 998.
3. Calculate the sum of all three-digit numbers within the given range.
- We can use the sum formula for arithmetic series, which is (n/2)(first term + last term), where n represents the number of terms. In this case, the first term is 103, the last term is 998, and we have to find the number of terms. We can calculate the number of terms by subtracting the first term from the last term and then adding 1 (since the terms are consecutive). So, the number of terms is (998 - 103) + 1 = 896.
4. Apply the formula to find the sum of all three-digit numbers.
- The sum of all three-digit numbers can be found using the formula (n/2)(first term + last term), where n represents the number of terms. So, the sum of all three-digit numbers that leave a remainder of three when divided by five is (896/2)(103 + 998).
5. Calculate the final answer.
- Evaluate the expression (896/2)(103 + 998) to find the sum of all three-digit numbers that meet the given condition. Implementing the necessary calculations, the sum of all three-digit numbers that leave a remainder of three when divided by five is 498,192.
Therefore, the sum of all three-digit numbers that leave a remainder of three when divided by five is 498,192.