Is The Special Property Satisfied?
I am looking for some special numbers satisfying a special property!
The special property is that the difference of the number of its odd factors and the number of tis even factors is an odd number.
So, how many such special numbers can be found from 1 to 1000, both inclusive?
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1 solution
Let A be the number of odd factors.
Let B be the number of even factors.
Because the difference of A and B is an odd number, either A is odd and B is even, or A is even and B is odd.
Nevertheless, the sum of the number of odd and even factors must be odd.
A positive integer has an odd number of factors only when it is a square number (the middle factor is all by itself, so it multiplies itself, resulting in a square number)
As 31 squared is 961 and 32 squared is 1024, the number of square numbers that can be found from 1 to 1000, both inclusive, is 31.
Hence, the answer is Between 25 to 50.