how many positive integers less than or equal to 100 have a prime factor that is greater than 4?
To find the number of positive integers less than or equal to 100, which have a prime factor greater than 4, we can use a step-by-step approach:
1. Determine the prime numbers less than 100.
- Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. The prime numbers less than 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
2. Identify the prime numbers greater than 4.
- Among the prime numbers less than 100, the prime numbers greater than 4 are: 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
3. Determine the positive integers less than or equal to 100 that have a prime factor greater than 4.
- We can divide each positive integer less than or equal to 100 by the prime numbers greater than 4 and check if the remainder is 0. If the remainder is 0, it means that the number has a prime factor greater than 4. Count these numbers.
Let's go through this step-by-step process to find the answer:
1. Determine the prime numbers less than 100: There are 25 prime numbers less than 100.
2. Identify the prime numbers greater than 4: There are 22 prime numbers greater than 4.
3. Determine the positive integers less than or equal to 100 that have a prime factor greater than 4: We divide each number from 1 to 100 by the prime numbers greater than 4 and count how many have a remainder of 0.
Here are the positive integers less than or equal to 100 that have a prime factor greater than 4:
10, 12, 15, 16, 18, 20, 21, 24, 25, 26, 27, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 51, 54, 55, 56, 57, 60, 63, 64, 65, 66, 69, 70, 72, 75, 77, 78, 80, 81, 84, 85, 86, 87, 88, 90, 91, 92, 93, 95, 96, 98, 99, and 100.
There are a total of 55 positive integers less than or equal to 100 that have a prime factor greater than 4.