Question

Петя задумал четыре различные цифры, не равные 0. Затем он всеми способами составил из этих цифр четырехзначные числа без повторяющихся цифр. Сумма всех этих чисел оказалась равна 173316. Какие 4 цифры задумал Петя, если известно, что среди них нет цифры 8?

Answer 1

Ответ:

Пошаговое объяснение:

так как сумма чисел очень большая для всех комбинаций четырёхзначных чисел , то начнем с самой большой без цифры 8

9765 сложив все комбинации получим сумму чуть больше 173316

,тогда уменьшим эту комбинацию минимум - получим 9764 , четырехзначных чисел выйдет 24 комбинации

сложив все комбинации с цифрами 9 , 7 , 6 , 4 - то получим сумму равную 173316

Ответ :Петя задумал 9 , 7 , 6 , 4

Answer 2

Problem Analysis

In this problem, Petya has chosen four different digits and used them to create four-digit numbers without repeating any digits. The sum of all these numbers is 173316. We need to determine which four digits Petya chose, knowing that the digit 8 is not among them.

Solution

Let's break down the problem step by step to find the solution.

1. Petya chose four different digits, not including 0 or 8. 2. Petya created four-digit numbers without repeating any digits. 3. The sum of all these numbers is 173316.

To solve this problem, we can use a systematic approach. Let's consider the possible values for each digit.

Possible Values for the Thousands Digit

Since the sum of all the four-digit numbers is 173316, the sum of the thousands digits must be a multiple of 1000. The only possible value for the thousands digit is 1.

Possible Values for the Hundreds Digit

The sum of the hundreds digits must be a multiple of 100. Let's consider the possible values for the hundreds digit:

- If the thousands digit is 1, the sum of the hundreds digits must be 73316. - Since the digit 8 is not among the chosen digits, the sum of the hundreds digits cannot be a multiple of 800. - Therefore, the sum of the hundreds digits must be a multiple of 100, but not a multiple of 800.

Let's consider the possible values for the hundreds digit:

- 2: If the hundreds digit is 2, the sum of the tens and units digits must be 3316. - 3: If the hundreds digit is 3, the sum of the tens and units digits must be 2316. - 4: If the hundreds digit is 4, the sum of the tens and units digits must be 1316. - 5: If the hundreds digit is 5, the sum of the tens and units digits must be 316. - 6: If the hundreds digit is 6, the sum of the tens and units digits must be 16.

Possible Values for the Tens and Units Digits

Let's consider the possible values for the tens and units digits based on the possible values for the hundreds digit:

- If the hundreds digit is 2, the sum of the tens and units digits must be 3316. - Possible values for the tens and units digits: 1 and 6. - If the hundreds digit is 3, the sum of the tens and units digits must be 2316. - Possible values for the tens and units digits: 1 and 5. - If the hundreds digit is 4, the sum of the tens and units digits must be 1316. - Possible values for the tens and units digits: 1 and 3. - If the hundreds digit is 5, the sum of the tens and units digits must be 316. - Possible values for the tens and units digits: 1 and 6. - If the hundreds digit is 6, the sum of the tens and units digits must be 16. - Possible values for the tens and units digits: 1 and 6.

Final Solution

Based on the analysis above, the possible combinations of the four digits Petya chose are:

1. Thousands digit: 1 2. Hundreds digit: 2, Tens digit: 1, Units digit: 6 3. Hundreds digit: 3, Tens digit: 1, Units digit: 5 4. Hundreds digit: 4, Tens digit: 1, Units digit: 3 5. Hundreds digit: 5, Tens digit: 1, Units digit: 6 6. Hundreds digit: 6, Tens digit: 1, Units digit: 6

Therefore, the four digits Petya chose are 1, 2, 1, and 6.

Conclusion

Petya chose the digits 1, 2, 1, and 6, and created four-digit numbers without repeating any digits. The sum of all these numbers is 173316.