How to Find the Product of Divisors from Prime Factorization

The product of all positive divisors of a number can be found directly from its prime factorization. There is no need to list and multiply every divisor separately. The main relationship uses the number itself and its total divisor count.

Product of Divisors Formula

If a positive integer n has τ(n) positive divisors, then the product of those divisors is:

P(n) = n^τ(n)/2

For a perfect square, the same result can be written without a fractional-looking exponent:

P(n) = n^{(τ(n) − 1)/2} × √n

Starting with the Prime Factorization

Suppose the prime factorization of n is:

Prime Factor Form

n = p₁^a₁ × p₂^a₂ × ⋯ × p_k^a_k

Each positive divisor is formed by choosing an exponent for every prime. For the prime p_i, the exponent may range from 0 through a_i. This gives the divisor-count formula:

Number of Positive Divisors

τ(n) = (a₁ + 1)(a₂ + 1)⋯(a_k + 1)

Once τ(n) is known, it can be substituted into the product formula. This connects prime factorization, the divisor count, and the product of divisors in one calculation.

Why the Formula Works

Positive divisors can be paired so that the product of each pair is n. A divisor d pairs with n / d:

1 and n
d and n / d
another divisor and its matching factor

If n has an even number of divisors, there are τ(n) / 2 pairs. Every pair has product n, so multiplying all pairs gives:

Divisor Pair Argument

n × n × ⋯ × n = n^τ(n)/2

A perfect square behaves slightly differently because √n pairs with itself. Its divisor count is odd. The remaining divisors form (τ(n) − 1) / 2 ordinary pairs, while the square root appears once.

Perfect Square Note

A positive integer has an odd number of positive divisors exactly when it is a perfect square. The unpaired divisor is its positive square root.

Example: Product of the Divisors of 72

The prime factorization of 72 is:

Factorization of 72

72 = 2³ × 3²

The number of positive divisors is determined from the exponents:

τ(72) = (3 + 1)(2 + 1) = 4 × 3 = 12

Since 72 is not a perfect square, its 12 divisors form six pairs. The product is therefore:

P(72) = 72^12/2 = 72⁶ = 139,314,069,504

Prime Factorization

2³ × 3²

Divisor Count

12 positive divisors

Number of Pairs

12 / 2 = 6 pairs

Divisor Product

72⁶ = 139,314,069,504

Example with a Perfect Square: 36

The prime factorization of 36 is:

36 = 2² × 3²

Its divisor count is:

τ(36) = (2 + 1)(2 + 1) = 9

Because 36 is a perfect square, the divisor 6 pairs with itself. The other eight divisors form four pairs:

P(36) = 36^{(9 − 1)/2} × √36

P(36) = 36⁴ × 6 = 10,077,696

The compact formula gives the same value:

P(36) = 36^9/2 = 10,077,696

Direct Formula from the Prime Exponents

The product may also be written entirely in terms of the prime factors. For

n = p₁^a₁p₂^a₂⋯p_k^a_k,

the total exponent of each prime p_i across all divisors is a_iτ(n) / 2. This gives:

Prime-Exponent Formula

P(n) = p₁^a₁τ(n)/2 × p₂^a₂τ(n)/2 × ⋯ × p_k^a_kτ(n)/2

This form may appear to create half-integer exponents when τ(n) is odd. That does not cause a problem. An odd divisor count means that n is a perfect square, so every prime exponent a_i is even. Each product a_iτ(n) / 2 is therefore an integer.

Non-Square and Perfect-Square Cases

The divisor product formula written for the two possible divisor-pair structures.
Case	Divisor Count	Pair Structure	Product Formula
Not a perfect square	Even	Every divisor has a different matching divisor	n^τ(n)/2
Perfect square	Odd	√n is paired with itself	n^{(τ(n) − 1)/2} × √n

Using the Method with Any Prime Factorization

Identify the Exponents

In the factorization of n, record the exponent attached to each distinct prime factor.

Calculate τ(n)

Add 1 to every prime exponent and multiply the resulting values.

Recognize a Square

If every prime exponent is even, n is a perfect square and τ(n) is odd.

Apply the Product Formula

Use n^τ(n)/2, or use the square-specific form when an integer exponent is easier to read.

Common Errors

Product Is Not the Divisor Sum

The divisor product function multiplies all positive divisors. It is different from the sum-of-divisors function, which adds them.

Only Positive Divisors Are Included

The standard formula concerns positive divisors. Including negative divisors changes the number of factors and may change the sign of the product.

Do Not Count the Square Root Twice

For a perfect square, √n is one divisor rather than two separate divisors. It is the reason the total divisor count is odd.

Related Divisor Concepts

Frequently Asked Questions

What is the formula for the product of all positive divisors?

If n has τ(n) positive divisors, their product is n^τ(n)/2. For a perfect square, it may also be written as n^{(τ(n) − 1)/2} × √n.

How is the divisor count found from prime factorization?

For n = p₁^a₁⋯p_k^a_k, the number of positive divisors is (a₁ + 1)⋯(a_k + 1).

Why are divisors paired when finding their product?

For every positive divisor d of n, the number n / d is also a divisor. Their product is n, allowing the full divisor product to be expressed as a power of n.