Find the secret code
You are interested in an organization and want to join it. But the organization takes only the best of all the applicants. There are applicants including you. The examiner takes every one in a giant room of push switches(every switch has a counter under it). Every time a switch is pushed the counter under it counts it. The rule is simple, the first man pushes all the switches, the second man pushes every other switch ( ), the third man pushes every third switch ( ) and so on till the last man. The switches which will have a score of 3 are used in the secret code. You have to find the numbers and write them in a string and submit it. Suppose the numbers are and then the code will be ' '. Find the code.
The answer is 492549121.
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1 solution
If we try out for 150 people by hand its going to take a lot of effort and time. Lets start looking for smaller groups.
If the number of people is equal to the switches, we can see some patterns after a few tries. Like every switch is hit when one of its factor number man is pushing switches. So ultimately we have to find which numbers have 3 factors.
It is obvious that factors come in pairs except for square numbers like 4, 9, 16, 25... because every factor of a square number has a pair except for its square root. As 3 is an odd number, only square numbers are suspects.
But every square number does not have 3 factors. For example 16 has 5 factors (1,2,4,8,16). But 4, 9, 25, 49 has 3 factors each. The thing they have in common is they are the squares of prime numbers. Such numbers in the given boundary are 4, 9, 25, 49, 121.
So the code is 492549121.