Malika wrote on the board in a row all integers from 9 to 2023.
Talgat came and replaced each of these numberes with the sum of its
digits. Then Vlad came and did the same with the resulting numbers.
This continued until 2015 singe-digit numbers remain on the board.
What is the sum of all the remaining numbers?
Ответы на вопрос
Let's calculate the sum of the remaining single-digit numbers after Malika, Talgat, and Vlad's operations.
First, we need to find a pattern for the numbers as they are transformed.
The sum of the digits of 9 is 9.
The sum of the digits of 10 is 1.
The sum of the digits of 11 is 2.
...
The sum of the digits of 19 is 10, which becomes 1.
The sum of the digits of 20 is 2.
The sum of the digits of 21 is 3.
...
The sum of the digits of 28 is 10, which becomes 1.
As we can see, every number from 10 to 19, as well as every number ending with 10, has a digit sum of 1. Similarly, every number from 20 to 29, as well as every number ending with 20, has a digit sum of 2. This pattern continues, and we can conclude that:
Numbers with a digit sum of 1 remain as 1 after the operation.
Numbers with a digit sum of 2 remain as 2 after the operation.
...
Numbers with a digit sum of 9 remain as 9 after the operation.
Now, let's find out which numbers have a digit sum of 1 to 9 within the range from 9 to 2023:
1, 10-19, 100-199, 1000-1999
Counting these numbers:
There is 1 number with a digit sum of 1 (which is 1 itself).
There are 10 numbers with a digit sum of 2 (from 10 to 19).
There are 100 numbers with a digit sum of 3 (from 100 to 199).
There are 1000 numbers with a digit sum of 4 (from 1000 to 1999).
For digit sums from 5 to 9, there are no numbers within the given range that meet these criteria.
Now, let's calculate the sum of these numbers:
1 * 1 + 10 * 2 + 100 * 3 + 1000 * 4 = 1 + 20 + 300 + 4000 = 4321
So, the sum of all the remaining single-digit numbers on the board after Malika, Talgat, and Vlad's operations is 4321.
Ответ:, the sum of all the remaining single-digit numbers on the board after Malika, Talgat, and Vlad's operations is 4321