Question

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?

Answer 1

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.

Answer 2

Ответ:, the sum of all the remaining single-digit numbers on the board after Malika, Talgat, and Vlad's operations is 4321

Answer 3

To solve this problem, we need to find the sum of all the remaining single-digit numbers after Malika, Talgat, and Vlad have replaced the original numbers with the sum of their digits.

First, let's consider the numbers from 9 to 2023. We notice that any number greater than 9 can be reduced to a single-digit number by summing its digits repeatedly. For example, 15 can be reduced to 1 + 5 = 6.

Since there are 2015 single-digit numbers remaining, we need to find the sum of these numbers.

The single-digit numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The sum of these numbers is 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45.

Therefore, the sum of all the remaining numbers is 45.

Answer 4

First, let's find the sum of all the remaining single-digit numbers on the board after Vlad's turn.

To do this, we'll start with the numbers from 9 to 2023 and replace each of them with the sum of their digits. Then, we'll repeat this process until only single-digit numbers remain.

Starting with 9, the sum of its digits is 9. For 10, the sum of its digits is 1. And so on, until 2023, where the sum of its digits is 7.

Now, let's continue replacing each of these resulting numbers with the sum of their digits until we're left with only single-digit numbers.

After repeating this process, we find that the remaining single-digit numbers are: 1, 2, 3, 4, 5, 6, 7, 8, and 9.

The sum of these numbers is 45.

So, the sum of all the remaining numbers on the board after Vlad's turn is 45.