Question

Определите какой цифрой заканчивается разность 2017^999-2016^777 ^ - степень

Answer 1

Я уже отвечал. Обозначим "оканчивается на ту же цифру" как =.
Имеют значение только последние цифры. 
2017^999 - 2016^777 = 7^999 - 6^777
7^4 = 2401 - кончается на 1, поэтому, если n кратно 4, то 7^n = 1.
7^999 = 7^996*7^3 = 1*343 = 343
6 в любой степени кончается на 6. 
6^777 = 6
Получаем
2017^999 - 2016^777 = 7^999 - 6^777 = 343 - 6 = 337
Это число кончается на 7.

Answer 2

Determining the Ending Digit of the Difference 2017^999 - 2016^777

To determine the ending digit of the difference 2017^999 - 2016^777, we can use the concept of modular arithmetic. The ending digit of a number in base 10 is the remainder when divided by 10. Therefore, we can find the remainder of the difference 2017^999 - 2016^777 when divided by 10 to determine its ending digit.

First, let's calculate the individual values of 2017^999 and 2016^777.

Calculating 2017^999: The value of 2017^999 is extremely large, but we can find its ending digit by using modular arithmetic. We can use the property that the ending digit of a number raised to a power repeats in a cycle. For example, the ending digit of 7^1 is 7, 7^2 is 9, 7^3 is 3, and so on. This creates a cycle of ending digits: 7, 9, 3, 1. Therefore, to find the ending digit of 2017^999, we can find the remainder of 999 when divided by 4, and use that to determine the ending digit.

Calculating 2016^777: Similarly, we can find the ending digit of 2016^777 using modular arithmetic. The ending digit of 6^1 is 6, 6^2 is 6, 6^3 is 6, and so on. This means the ending digit of 2016^777 is 6.

Now, let's find the difference 2017^999 - 2016^777 and determine its ending digit.

Finding the Ending Digit of the Difference: We can subtract the ending digit of 2016^777 from the ending digit of 2017^999 to find the ending digit of the difference.

Ending Digit of 2017^999: The remainder of 999 when divided by 4 is 3. Therefore, the ending digit of 2017^999 is the same as the ending digit of 7^3, which is 3.

Ending Digit of 2016^777: The ending digit of 2016^777 is 6.

Difference and Ending Digit: The difference 2017^999 - 2016^777 is 3 - 6 = -3. However, since we are interested in the ending digit, we take the positive remainder when divided by 10, which is 7.

Therefore, the ending digit of the difference 2017^999 - 2016^777 is 7.