Input a number to check if it's a powerful number, or define a range to generate all powerful numbers within it.
Powerful Number Check or Generate
Result
Result
What is a Powerful Number?
A powerful number is a positive integer \( n \) such that for every prime factor of \( n \), the square of that prime is also a divisor of \( n \). In mathematical terms, if \( n \) can be factored as \( p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k} \), then each \( p_i^2 \) must divide \( n \), and all \( e_i \geq 2 \).
How to Determine a Powerful Number
Perform Prime Factorization: Identify the prime factors of \( n \) and their respective exponents.
Check the Exponents: Verify that the exponent of every prime factor is at least 2.
Conclusion: If all exponents meet the condition, \( n \) is a powerful number; otherwise, it is not.
Examples
Example 1: Check if 36 is a Powerful Number
Solution:
1. Prime Factorization:
\( 36 = 2^2 \times 3^2 \)
2. Check:
All prime factors have exponents greater than or equal to 2.
Result:
36 is a powerful number.
Example 2: Check if 2025 is a Powerful Number
Solution:
1. Prime Factorization:
\( 2025 = 3^4 \times 5^2 \)
2. Check:
All prime factors have exponents greater than or equal to 2.