The following operation is repeated for a certain integer repeatedly.
<Operation>
It doubles.
However,
When the doubled number is 150 or more, 100 is subtracted from this doubled number.
When the doubled numbers is 101 or more and 149 or less, 50 is subtracted from this doubled number.
When the doubled number is 100 or less, it leaves in this doubled number.
For example, when this operation is repeated 4 times starting with 36, the integer acquired is 72, 94, 88, and 76 at order.
(1) When you repeat this operation 4 times, find the number of integers that the result becomes 60.
(2) When you repeat this operation 101 times, find the least number among the integers that the result becomes 60.
It doubles.
However,
When the doubled number is 150 or more, 100 is subtracted from this doubled number.
When the doubled numbers is 101 or more and 149 or less, 50 is subtracted from this doubled number.
When the doubled number is 100 or less, it leaves in this doubled number.
For example, when this operation is repeated 4 times starting with 36, the integer acquired is 72, 94, 88, and 76 at order.
(1) When you repeat this operation 4 times, find the number of integers that the result becomes 60.
(2) When you repeat this operation 101 times, find the least number among the integers that the result becomes 60.
Answer
Solution
(1) It goes back in good order and thinks.
Since it is a number which doubles and becomes 160, 110, and 60, it is 80, 55, or 30 that becomes 60 after the 4th operation.
Since there is no number which doubles and becomes 55, 55 is not suitable as a number after the 3rd operation.
55 is excepted.
It is 90 and 40 that becomes 80 or 30 after the 3rd operation.
It is 70 and 20 that becomes 90 or 40 after the 2nd operation.
Furthermore, since it is 85, 60, 35, and 10 that becomes 70 or 20 after the 1st operation.
Therefore the number of suitable integer is four.
(2) Following (1), when the number becomes 60 by five operations is picked up, they are four numbers, 80, 55, 30, and 5.
When the number becomes 60 by six operations is also picked up, it is 90, 65, 40, and 15.
The result of (1) shows 60 turning into 60 by four operations and being set to 60 by repetition of four operations.
Then, according to the calculation of 101 / 4 = 25 with remainder 1, the number suitable as an answer is in numbers which becomes 60 by one or five operation.
Since the numbers becomes 60 by 1 time or 5 times are 80, 55, 30, and 5, the least number in these is 5.
(1) Four (10, 35, 60, 85)
(2) 5
(2) 5
(1) It goes back in good order and thinks.
Since it is a number which doubles and becomes 160, 110, and 60, it is 80, 55, or 30 that becomes 60 after the 4th operation.
Since there is no number which doubles and becomes 55, 55 is not suitable as a number after the 3rd operation.
55 is excepted.
It is 90 and 40 that becomes 80 or 30 after the 3rd operation.
It is 70 and 20 that becomes 90 or 40 after the 2nd operation.
Furthermore, since it is 85, 60, 35, and 10 that becomes 70 or 20 after the 1st operation.
Therefore the number of suitable integer is four.
(2) Following (1), when the number becomes 60 by five operations is picked up, they are four numbers, 80, 55, 30, and 5.
When the number becomes 60 by six operations is also picked up, it is 90, 65, 40, and 15.
The result of (1) shows 60 turning into 60 by four operations and being set to 60 by repetition of four operations.
Then, according to the calculation of 101 / 4 = 25 with remainder 1, the number suitable as an answer is in numbers which becomes 60 by one or five operation.
Since the numbers becomes 60 by 1 time or 5 times are 80, 55, 30, and 5, the least number in these is 5.
