The Number of Positive Divisors of an Integer Examples 1

Recall from The Number of Positive Divisors of an Integer page that if $n \in \mathbb{Z}$ then the number of positive divisors of $n$ is denoted $d(n)$ and is given explicitly by:

(1)

\begin{align} \quad d(n) = \sum_{d \mid n}_{d > 0} 1 \end{align}

Most importantly, we noted that:

• If $p$ is prime then $d(p) = 2$.

• If $p$ is prime and $k \in \mathbb{N}$ then $d(p^k) = k + 1$.

• If $p$ and $q$ are prime then $d(pq) = d(p)d(q)$.

We will now look at some examples of computing regarding the number of positive divisors of an integer. More examples can be found on The Number of Positive Divisors of an Integer Examples 2 page.

Example 1

Calculate $d(12322)$.

We first note that the prime power decomposition of $12322 = 2 \cdot 61 \cdot 101$. Hence:

(2)

\begin{align} d(12322) & = d(2) d(61) d(101) \\ d(12322) & = (2)(2)(2) \\ d(12322) & = 8 \end{align}

Example 2

Calculate $d(88888)$.

The prime power decomposition of $88888 = 2^3 \cdot 41 \cdot 271$. Hence:

(3)

\begin{align} d(88888) &= d(2^3) d(41) d(271) \\ d(88888) &= (4)(2)(2) \\ d(88888) &= 16 \end{align}

Example 3

Calculate $d(500000)$.

The prime power decomposition of $500000 = 2^5 \cdot 5^6$. Hence:

(4)

\begin{align} d(500000) &= d(2^5) d(5^6) \\ d(500000) &= (6)(7) \\ d(500000) &= 42 \end{align}

Example 4

Calculate $d(32930020)$.

The prime power decomposition of $32930020 = 2^2 \cdot 5 \cdot 17 \cdot 23 \cdot 4211$. Hence:

(5)

\begin{align} d(32930020) & = d(2^2) d(5) d(17) d(23) d(4211) \\ d(32930020) & = (3)(2)(2)(2)(2) \\ d(32930020) & = 48 \end{align}

Example 5

Calculate $d(9876543210)$.

The prime power decomposition of $9876543210 = 2 \cdot 3^2 \cdot 5 \cdot 17^2 \cdot 379721$. Hence:

(6)

\begin{align} d(9876543210) & = d(2) d(3^2) d(5) d(17^2) d(379721) \\ d(9876543210) & = (2)(3)(2)(3)(2) \\ d(9876543210) & = 72 \end{align}