Finding the Number of Factors of a Number

The formula

Write a number as a product of prime powers, $N = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ . Every factor of $N$ is built by independently choosing an exponent from $0$ to $a_i$ for each prime, so the total count of factors (including 1 and $N$ itself) is:

(a_1+1)(a_2+1)\cdots(a_k+1)

Worked example

How many divisors does $9600$ have, including 1 and 9600 itself?

Solution: Prime factorize first:

9600 = 2^7 \times 3^1 \times 5^2

Add 1 to each exponent and multiply:

(7+1)(1+1)(2+1) = 8 \times 2 \times 3 = 48

Worked example: already factored

How many positive divisors does $2^5 \cdot 3^6 \cdot 7^3$ have?

Solution: The prime factorization is already given, so apply the formula directly:

(5+1)(6+1)(3+1) = 6 \times 7 \times 4 = 168

Finding the Number of Factors of a Number

The formula

(a_1+1)(a_2+1)\cdots(a_k+1)

Worked example

How many divisors does $9600$ have, including 1 and 9600 itself?

Solution: Prime factorize first:

9600 = 2^7 \times 3^1 \times 5^2

Add 1 to each exponent and multiply:

(7+1)(1+1)(2+1) = 8 \times 2 \times 3 = 48

Worked example: already factored

How many positive divisors does $2^5 \cdot 3^6 \cdot 7^3$ have?

Solution: The prime factorization is already given, so apply the formula directly:

(5+1)(6+1)(3+1) = 6 \times 7 \times 4 = 168

Finding the Number of Factors of a Number

The formula

Worked example

Worked example: already factored

Continue learning

Finding the Number of Factors of a Number

The formula

Worked example

Worked example: already factored

Continue learning