Question

Сколько существует таких натуральных чисел AA, что среди чисел AA, A+10A+10 и A+20A+20 ровно два четырехзначных?

Answer 1

А=80,81,82...87,88,89
8080 дает 8080, 9090, 100100,
8181 дает 8181,9191. 101101 ....
....
8989 дает 8989,9999,109109
-всего 10шт

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Answer 2

Count of Natural Numbers

To find the count of natural numbers A such that among the numbers A, 10A + 10, and A + 20, there are exactly two four-digit numbers, we can approach the problem systematically.

First, let's consider the number A itself. Since A is a natural number, it must be greater than or equal to 1000 to ensure that 10A + 10 and A + 20 can potentially be four-digit numbers.

Next, we need to determine the conditions under which exactly two of the numbers A, 10A + 10, and A + 20 are four-digit numbers.

Analysis and Solution

To ensure that exactly two of the numbers A, 10A + 10, and A + 20 are four-digit numbers, we can consider the following cases:

1. If A is a four-digit number, then A, 10A + 10, and A + 20 are all four-digit numbers. 2. If A is a three-digit number, then either 10A + 10 or A + 20 (but not both) will be a four-digit number. 3. If A is a two-digit number, then A + 20 will be a four-digit number.

By considering these cases, we can calculate the count of natural numbers A that satisfy the given conditions.

Calculation

Let's calculate the count of natural numbers A based on the above cases:

1. For A as a four-digit number: There are 9000 four-digit numbers from 1000 to 9999. 2. For A as a three-digit number: There are 900 three-digit numbers from 100 to 999. 3. For A as a two-digit number: There are 90 two-digit numbers from 10 to 99.

Adding these counts together, we get the total count of natural numbers A that satisfy the given conditions.

Result

The count of natural numbers A that satisfy the given conditions is the sum of the counts from the three cases mentioned above.

Note: The specific count of natural numbers A can be calculated by summing the counts from each case as described above.

Let me know if you need further assistance with this calculation!