We start with a two-digit positive integer and construct a sequence of two-digit numbers as follows. Let the current number be x. If 2x is less than 100, then the next number in the sequence is 2x. Otherwise, the next number in the sequence is 2x − 100.
A number is said to be good if we can start with the number and get back to the same number later in the sequence. A number that is not good is said to be bad.
For example, 20 is a good number, because the sequence starting with 20 is 20, 40, 80, 60, 20. So, after four steps, we get back to 20. However 10 is bad because starting from 10 we get the sequence 10, 20, 40, 80, 60, 20, . . . in which 10 never reappears. What is the common property that is shared by the set of good numbers?
- Zonal Informatics Olympiad 2003
Answers
Answer:
Step-by-step explanation:
Here,we are observing that a number x has to be a multiple of 4,in order to satisfy the requirements of a good number.
Any other multiple won't satisfy.
As we are multiplying the number with 2,we are either getting 2x or 2x-100,as per condition.
But,that 2x and 2x-100 will also be a multiple of 4.
A series is obtained such that it purely increases first and then decreases back to the original number x itself.
The above is the main common property.
Take x=16(Multiple of 4)
16,32,64,28,56,12,24,48,96,92,84,68,
36,72,44,88,76,52,4,16,....
Hence,we get 16 again,so it is good.
Secondly,no odd number is good.All odd numbers are bad.
Take x=15
According to series,
15,30,60,20,40,80,60,........
We don't get 15 again, so it is bad.