What is the sum of all positive integers N≤1000 such that N equals 13 times the digit sum of N?

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This is a Brilliant question. Please do not answer.

To find the sum of all positive integers N≤1000 such that N equals 13 times the digit sum of N, we need to follow these steps:

1. Start with the smallest value for N, which is 1.
2. Calculate the digit sum of N by adding up its individual digits. For example, the digit sum of 1 is 1, and the digit sum of 29 is 2+9=11.
3. Multiply the digit sum of N by 13.
4. Compare the result of step 3 with N. If they are equal, add N to the running sum. If not, move on to the next value of N.
5. Repeat steps 2-4 until you reach N=1000.
6. Sum up all the values of N from step 4 to get the final answer.

Let's go through the process step by step:

Start with N=1:
The digit sum of 1 is 1.
Multiplying the digit sum by 13: 1 * 13 = 13.
Since 13 is equal to N, we add it to the sum (running total).

Move on to N=2:
The digit sum of 2 is 2.
Multiplying the digit sum by 13: 2 * 13 = 26.
Since 26 is not equal to N, we move on to the next value.

Continue this process for N=3, 4, 5, and so on until N=1000. At each step, calculate the digit sum, multiply it by 13, and check if it is equal to N.

Finally, sum up all the values of N that were equal to 13 times their digit sum to get the desired answer.