Question

How many positive integers less than 500 have an odd number of factors?

Answer 1

Answer:

22

Step-by-step explanation:

Remark

Very nice little problem.

At first you think you need to go through the entire number system from 1 to 500. For example, you could do 12 as 1 2 3 4 6 12 So 12 does not add to the number of integers that have an odd number of factors.

But along the way if you check all the factors of the integers from 1 to 12, you notice something very odd. (No pun intended).

1 only has 1 factor which is odd (not even).

4 has 3 factors 1 2 4 which is odd.

9 has 3 factors 1 3 9 which is odd.

It turns out that there are 22 numbers from 1 to 500 that have an odd number of factors. They are the perfect squares between 1 and 500

Note

The factors of an integer include 1 and the integer itself

As a conclusion, try 36

1 2 3 4 6 9 12 18 36  There's 9 factors which is odd. So only the perfect squares have an odd number of factors.