Integral Divisors

This section covers formulas for analyzing the divisors of an integer. First, prime factorize the number $n$ as $n = p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{e_k}$.

• Number of Prime Divisors: The number of distinct prime factors ($k$).
• Number of Integral Divisors: $(e_1+1)(e_2+1)\cdots(e_k+1)$.
• Sum of Integral Divisors: $(\frac{p_1^{e_1+1}-1}{p_1-1}) \cdots (\frac{p_k^{e_k+1}-1}{p_k-1})$.
• Proper Divisors: All divisors except the number itself.
• Relatively Prime Integers less than n (Euler's Totient Function): The number of integers less than $n$ that are coprime to $n$ is $n(1-\frac{1}{p_1})\cdots(1-\frac{1}{p_k})$.