To find the number of divisors a number has, you first need to find its prime factorization. Then, use the exponents in the prime factorization to calculate the number of divisors.
The Formula
The number of divisors, d(n), of a natural number n can be determined as follows:
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Prime Factorization: Express n as a product of its prime factors:
n = p₁ᵃ¹ p₂ᵃ² p₃ᵃ³ ... * pₖᵃᵏ
where p₁, p₂, p₃, ..., pₖ are distinct prime numbers and a₁, a₂, a₃, ..., aₖ are positive integers.
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Divisor Count: The number of divisors is then calculated by:
d(n) = (a₁ + 1)(a₂ + 1)(a₃ + 1)...(aₖ + 1)
Explanation
The formula works because any divisor of n must be of the form:
p₁ᵇ¹ p₂ᵇ² p₃ᵇ³ ... * pₖᵇᵏ
where 0 ≤ b₁ ≤ a₁, 0 ≤ b₂ ≤ a₂, 0 ≤ b₃ ≤ a₃, ..., 0 ≤ bₖ ≤ aₖ.
For each prime factor pᵢ, there are (aᵢ + 1) choices for the exponent bᵢ (ranging from 0 to aᵢ). Since the choices for each prime factor are independent, the total number of divisors is the product of the number of choices for each exponent.
Example
Let's find the number of divisors of 360.
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Prime Factorization: 360 = 2³ 3² 5¹
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Divisor Count: d(360) = (3 + 1)(2 + 1)(1 + 1) = 4 3 2 = 24
Therefore, 360 has 24 divisors.
Summary
To determine the number of divisors a number has, perform prime factorization and then apply the formula: d(n) = (a₁ + 1)(a₂ + 1)(a₃ + 1)...(aₖ + 1), where a₁, a₂, a₃, ..., aₖ are the exponents in the prime factorization.
