For how many odd positive integers n<1000 does the number of positive divisors of n divide n?

30

5 :p

To find the number of odd positive integers n<1000 for which the number of positive divisors of n divides n, we can follow these steps:

Step 1: List down all the odd positive integers less than 1000.
- Starting from 1, increment by 2 until 999.
- This gives us a list of all odd positive integers less than 1000.

Step 2: For each odd number, find the number of its positive divisors.
- One way to do this is to find its prime factorization.
- For example, for the number 75, the prime factorization is 3^1 * 5^2.
- The number of divisors can be calculated by adding 1 to each of the exponents and multiplying them together: (1+1) * (2+1) = 6.
- So, the number of positive divisors for 75 is 6.

Step 3: Check if the number of divisors divides the number itself.
- For example, for the number 75 with 6 positive divisors, we check if 6 divides 75.
- If the remainder of the division is zero, then the number of divisors divides the number.
- If the remainder is not zero, then the number of divisors does not divide the number.

Step 4: Count the number of odd positive integers that satisfy the condition.
- Iterate through all the odd positive integers and perform steps 2 and 3.
- Keep count of the numbers that satisfy the condition.

Following these steps, you can find the answer to the given problem.