What is the solution for number of positive integers less than 1000 with exactly 15 factors?

Question

this question is at the end of practice problems 2 video in Number system, the answer said was 4, i didnt understand why the answer will be only 4, its more from what im solving, please clarify for the same

Kunal · Accepted Answer

To determine the number of positive integers less than 1000 with exactly 15 factors, we need to use the properties of divisors.Specifically, we use the formula for the number of divisors of an integer n. If n has a prime factorization of the form:n = p1^e1 * p2^e2 * ... * pk^ek
then the number of divisors d(n) of n is given by:d(n) = (e1 + 1)(e2 + 1) ... (ek + 1)
We need d(n) = 15. The number 15 can be factored in the following ways:1. 15 = 15 * 12. 15 = 5 * 33. 15 = 3 * 5
Let's explore each case to find the positive integers less than 1000 with exactly 15 factors.
Case 1: 15 = 15 * 1In this case, n would be of the form p1^14 where p1 is a prime number.- 2^14 = 16384 (not less than 1000)- 3^14 = 4782969 (not less than 1000)- Higher primes will yield even larger results.
There are no integers in this form that are less than 1000.
Case 2: 15 = 5 * 3In this case, n would be of the form p1^4 * p2^2 where p1 and p2 are distinct prime numbers.- Smallest primes are 2 and 3:  n = 2^4 * 3^2 = 16 * 9 = 144 (less than 1000)- Next smallest primes are 2 and 5:  n = 2^4 * 5^2 = 16 * 25 = 400 (less than 1000)- Next smallest primes are 3 and 5:  n = 3^4 * 5^2 = 81 * 25 = 2025 (not less than 1000)
Only 144 and 400 fit this form and are less than 1000.
Case 3: 15 = 3 * 5In this case, n would be of the form p1^2 * p2^4 where p1 and p2 are distinct prime numbers.- Smallest primes are 2 and 3:  n = 2^2 * 3^4 = 4 * 81 = 324 (less than 1000)- Next smallest primes are 2 and 5:  n = 2^2 * 5^4 = 4 * 625 = 2500 (not less than 1000)- Next smallest primes are 3 and 5:  n = 3^2 * 5^4 = 9 * 625 = 5625 (not less than 1000)
Higher primes will yield even larger results. Thus, only 324 fits this form and is less than 1000.
Final Count:Combining the results from the valid cases, we have the integers:- 144- 400- 324
Thus, there are exactly 3 positive integers less than 1000 with exactly 15 factors.