Question

Одно из двух целых чисел уменьшили в 2021 раз, другое увеличили также в 2021 раз, при этом среднее арифметическое исходных чисел не изменилось. Докажите, что это среднее арифметическое делится на 1011 без остатка.?​

Answer 1

Ответ:

Пошаговое объяснение:незнаю

Answer 2

Problem Analysis

We are given two whole numbers that have been multiplied by different factors in the year 2021. Despite this, the average of the original numbers remains unchanged. We need to prove that this average is divisible by 1011 without a remainder.

Solution

Let's assume the two original numbers are x and y. We are told that one of the numbers was decreased by a factor of 2021, while the other was increased by the same factor. The average of the original numbers is denoted as A.

We can express the new numbers as follows: - The decreased number: x/2021 - The increased number: y * 2021

The new average, denoted as B, can be calculated as follows: B = (x/2021 + y * 2021) / 2

We are given that A = B, which means the average remains unchanged. Therefore, we can write the equation: (x/2021 + y * 2021) / 2 = (x + y) / 2

To prove that the average is divisible by 1011 without a remainder, we need to show that (x + y) is divisible by 1011.

Let's simplify the equation: (x/2021 + y * 2021) = (x + y)

Multiplying both sides by 2021: x + 2021^2 * y = 2021 * (x + y)

Rearranging the equation: x - 2021 * x = 2021^2 * y - 2021 * y

Simplifying further: -2020 * x = 2020 * 2021 * y

Dividing both sides by 2020: -x = 2021 * y

Rearranging the equation: x = -2021 * y

We can see that x is divisible by 2021, which means (x + y) is also divisible by 2021. Since 1011 is a divisor of 2021, we can conclude that (x + y) is divisible by 1011.

Therefore, the average of the original numbers is divisible by 1011 without a remainder.

Conclusion

We have proven that if one whole number is decreased by a factor of 2021 and the other is increased by the same factor, while the average of the original numbers remains unchanged, then the average is divisible by 1011 without a remainder.

Please let me know if I can help you with anything else.