The Sum of All Four-Digit Numbers Divisible by 3 and 5

In this problem, we are required to find the sum of all four-digit numbers that are divisible by both 3 and 5. To solve this, we will break down the problem into smaller steps and apply mathematical concepts to find the solution.

Step 1: Identifying Divisibility
To identify the numbers divisible by 3, we need to check if the sum of their digits is divisible by 3. Similarly, for divisibility by 5, the number must end with 0 or 5. Since we are looking for numbers that are divisible by both 3 and 5, they must satisfy both conditions.

Step 2: Finding the Range of Four-Digit Numbers
Four-digit numbers range from 1000 to 9999. So, we need to find the numbers within this range that are divisible by both 3 and 5.

Step 3: Finding the Divisible Numbers
Let's break this step into two parts:

3.1: Divisibility by 3
To find the numbers divisible by 3, we need to calculate the sum of their digits. The sum of digits from 1 to 9 is 45, which is divisible by 3. We can observe that each subsequent group of 10 consecutive numbers will have a sum that is a multiple of 3. Hence, we can divide the range of four-digit numbers into 10 groups, each with a sum of 45.

3.2: Divisibility by 5
Since the numbers divisible by 5 must end with 0 or 5, we can calculate the number of such digits in each group. In a group of 10 numbers, there will be 2 numbers ending with 0 or 5.

Step 4: Calculating the Sum
To calculate the sum of all the four-digit numbers divisible by both 3 and 5, we need to find the sum for each group and then multiply it by the number of groups.

4.1: Sum for Each Group
The sum of each group will consist of the sum of digits (45) multiplied by the number of digits ending with 0 or 5 (2).

4.2: Number of Groups
The number of groups can be calculated by dividing the total range of four-digit numbers (9000) by 10.

Step 5: Final Calculation
To find the sum of all four-digit numbers divisible by both 3 and 5, we multiply the sum for each group (45 * 2) by the number of groups (9000 / 10).

Step 6: Simplification
Performing the calculations, we have:
Sum = (45 * 2) * (9000 / 10)
= 90 * 900
= 81000

Hence, the sum of all four-digit numbers divisible by both 3 and 5 is 81,000.