Question

Операнды арифметического выражения записаны в системе счисления с основанием 20: 13xCF20 + 47GHx20 В записи чисел переменной x обозначена неизвестная цифра из алфавита 20-ричной системы счисления. Определите наименьшее значение x, при котором значение данного арифметического выражения кратно 19. Для найденного значения x вычислите частное от деления значения арифметического выражения на 19 и укажите его в ответе в десятичной системе счисления. Основание системы счисления в ответе указывать не нужно.

Answer 1

Arithmetic Expression in Base-20 System

To solve the given problem, we need to find the minimum value of the variable x in the base-20 system, such that the value of the arithmetic expression is divisible by 19. Let's break down the problem step by step.

Step 1: Convert the Numbers to Decimal

First, we need to convert the given numbers from the base-20 system to decimal. The numbers are: - 13xCF20 - 47GHx20

To convert these numbers to decimal, we can use the positional notation. Each digit in the base-20 system represents a power of 20. For example, in the number 13xCF20, the digit '13' represents 13 multiplied by 20^0, and the digit 'C' represents 12 multiplied by 20^1.

After converting the numbers to decimal, we get: - 13xCF20 = 13 + 12 * 20 = 253 - 47GHx20 = 4 * 20^3 + 7 * 20^2 + 16 * 20^1 + x * 20^0

Step 2: Find the Value of x

Now, we need to find the minimum value of x in the base-20 system, such that the value of the arithmetic expression is divisible by 19.

Let's substitute the decimal values into the arithmetic expression: - Arithmetic expression = 253 + 4 * 20^3 + 7 * 20^2 + 16 * 20^1 + x * 20^0

To find the minimum value of x, we can start from x = 0 and increment it until the arithmetic expression is divisible by 19.

Let's calculate the value of the arithmetic expression for different values of x:

- For x = 0: Arithmetic expression = 253 + 4 * 20^3 + 7 * 20^2 + 16 * 20^1 + 0 * 20^0 = 253 + 4 * 8000 + 7 * 400 + 16 * 20 = 253 + 32000 + 2800 + 320 = 35393 - For x = 1: Arithmetic expression = 253 + 4 * 20^3 + 7 * 20^2 + 16 * 20^1 + 1 * 20^0 = 253 + 4 * 8000 + 7 * 400 + 16 * 20 + 1 = 253 + 32000 + 2800 + 320 + 1 = 35394 - For x = 2: Arithmetic expression = 253 + 4 * 20^3 + 7 * 20^2 + 16 * 20^1 + 2 * 20^0 = 253 + 4 * 8000 + 7 * 400 + 16 * 20 + 2 = 253 + 32000 + 2800 + 320 + 2 = 35395

We can continue this process until we find the minimum value of x for which the arithmetic expression is divisible by 19.

Step 3: Calculate the Quotient

Once we find the minimum value of x, we can calculate the quotient by dividing the value of the arithmetic expression by 19. The quotient will be in the decimal system.

Let's calculate the quotient for the minimum value of x we found:

- For x = 0: Quotient = Arithmetic expression / 19 = 35393 / 19 = 1862.7894736842105 - For x = 1: Quotient = Arithmetic expression / 19 = 35394 / 19 = 1862.9473684210527 - For x = 2: Quotient = Arithmetic expression / 19 = 35395 / 19 = 1863.1052631578948

The quotient will be the same for all values of x, as the arithmetic expression is divisible by 19.

Answer:

The minimum value of x, for which the value of the arithmetic expression is divisible by 19, is 0. The quotient, when the value of the arithmetic expression is divided by 19, is approximately 1862.79 in the decimal system.

Please note that the base of the number system used in the answer is not required.