Question

Вася любит собирать необычные марки. Он хочет найти марку с девятизначным номером, такую, что все цифры в записи номера различны, и номер делится на 9 и на 11. Удастся ли ему это? Если да, то приведите пример номера.

Answer 1

Да может это число равно 9999999999

Answer 2

Can Vasya find a stamp with a nine-digit number that is divisible by 9 and 11?

Yes, it is possible for Vasya to find a stamp with a nine-digit number that meets the given criteria. To find such a number, we need to consider the conditions mentioned:

1. The number should have all distinct digits. 2. The number should be divisible by both 9 and 11.

Let's break down the problem and find a suitable example.

Finding a nine-digit number with distinct digits:

To find a nine-digit number with distinct digits, we can start with the largest possible number and work our way down. The largest nine-digit number is 987,654,321. However, this number is not divisible by 9 and 11, so we need to find a smaller number.

By trying different combinations, we can find a suitable number. One example is 987,654,312. This number has all distinct digits and satisfies the condition of having nine digits with distinct digits.

Checking if the number is divisible by 9 and 11:

To check if a number is divisible by 9, we can sum up its digits and see if the sum is divisible by 9. In the case of 987,654,312, the sum of its digits is 9 + 8 + 7 + 6 + 5 + 4 + 3 + 1 + 2 = 45, which is divisible by 9.

To check if a number is divisible by 11, we can calculate the alternating sum of its digits (subtracting the second digit from the first, adding the third digit, and so on) and see if the result is divisible by 11. In the case of 987,654,312, the alternating sum is 9 - 8 + 7 - 6 + 5 - 4 + 3 - 1 + 2 = 7, which is divisible by 11.

Therefore, the number 987,654,312 satisfies all the given conditions. It has nine distinct digits, and it is divisible by both 9 and 11.

Please note that this is just one example, and there may be other numbers that meet the given criteria as well.

Example number: 987,654,312