[ Decimal number system ] [ Divisibility of numbers. General properties ]

In a 100-digit number 12345678901234...7890, all digits standing at odd positions were erased; in the resulting 50-digit number, all digits standing at odd positions were erased again, and so on. The erasing continued until there was nothing left to erase. What was the last digit to be erased?
What characterizes the ordinal numbers of the digits that remain after the first deletion? And after the second?

## Solution

After the first deletion, only those digits will remain whose initial numbers are even, after the second, those whose initial numbers are divisible by 4, after the third - by 8, and so on. Before the last deletion, a digit will remain whose initial number is equal to the greatest possible power of 2, that is, 64. This digit is 4.

## Answer

## 4

Numbers greater than three natural numbers are written in a circle, the sum of which is 37. It is known that the sums of any three consecutive numbers are equal. What numbers are written in the circle?

## Hint

Try to consider two cases: a) the number of written numbers is not a multiple of 3; b) the number of written numbers is a multiple of 3.

## Solution

Since the sums of any three consecutive numbers written in a circle are equal to each other, every third number is equal to each other. Consider two cases: a) the number of written numbers is not a multiple of 3; b) the number of written numbers is a multiple of 3. In the first case, all numbers will be equal to each other, and in the second - the sum of them will be a multiple of 3. The second case is impossible since 37 is not divisible by 3. In the first case, the only possibility is to write 37 ones in a circle.

## Answer

Ones.