How many divisors does 9 factorial have?

Answer 1

9! = 9×8×7×6×5×4×3×2

Separate each of the above factors into prime and non-prime:

Prime

2

3

5

7

Non-prime

4

6

8

9

Perform a prime factorization of each of the non-prime factors:

4 = 22

6 = 3×2

8 = 23

9 = 32

Rewrite the number using these prime factorizations:

9! = 32 × 23 × 7 × (3×2) × 5 × 22 × 3 × 2

Group:

9! = 7 × 5 × 34 × 27

For any number with factors (greater than 1) of aA × bB × cC × ... × nN , it can be shown that the total number of divisors is (A+1)×(B+1)×(C+1)× ... × (N+1). This is because there are N+1 possible ways to divide out the factor n to create a unique divisor (n0 is also a factor). Using basic combinatorics, the total possible number of divisors is simply the products of all these possibilities for each prime factor.

Therefore, the number of divisors in 9! can be computed as follows:

(1+1)(1+1)(4+1)(7+1) = 160

Thus, 9 factorial has 160 divisors.