Final answer:
There are 10 positive integers less than 100 that are powerful.
Step-by-step explanation:
A positive integer is called powerful if every prime in its prime factorization is raised to a power greater than or equal to 2. To find the number of positive integers less than 100 that are powerful, we can check each integer from 1 to 100 and determine if its prime factorization contains any primes raised to a power less than 2.
Let's go through the numbers one by one:
- For 1, the only prime factor is 1 and it is not raised to a power greater than or equal to 2. So 1 is not powerful.
- For 2, the prime factorization is just 2. Since 2 is raised to the power of 1 (which is less than 2), 2 is not powerful.
- For 3, the prime factorization is just 3. Since 3 is raised to the power of 1 (which is less than 2), 3 is not powerful.
- ...
- ... (continue checking all numbers less than 100 in the same way)
- For 97, the prime factorization is just 97. Since 97 is raised to the power of 1 (which is less than 2), 97 is not powerful.
- For 98, the prime factorization is 2 x 7 x 7. Since 2 is raised to the power of 1 (which is less than 2), 98 is not powerful.
- For 99, the prime factorization is 3 x 3 x 11. Since 3 is raised to the power of 2 (which is greater than or equal to 2) and 11 is raised to the power of 1 (which is less than 2), 99 is not powerful.
By checking all the numbers less than 100 in this way, we find that there are 10 positive integers less than 100 that are powerful.