# Divisors of 15

The divisors of 15 are those numbers that completely divide 15 with the remainder zero. In this section, we will discuss about divisors of 15.

#### Highlights of Divisors of 15

• Divisors of 15: 1, 3, 5 and 15
• Negative divisors of 15: -1, -3, -5 and -15
• Prime divisors of 15: 3 and 5
• Number of divisors of 15: 4
• Sum of divisors of 15: 24
• Product of divisors of 15: 152

#### What are Divisors of 15

A number n is a divisor of 15 if $\frac{15}{n}$ is an integer. Note that if 15/n=m is an integer, then both m and n will be the divisors of 15.

To find the divisors of 15, we need to find the numbers n such that 15/n becomes an integer. We have:

 15/1=15 1, 15 are divisors of 15. 15/3=5 3, 5 are divisors of 15

No numbers other than 1, 3, 5 and 15 can divide 15. So we conclude that

 The divisors of 15 are: 1, 3, 5 and 15.

Thus, the total number of divisors of 15 is four.

#### Negative Divisors of 15

We know that if m is a divisor of a number, then -m is also a divisor of that number.

As the divisors of 15 are 1, 3, 5, and 15, we can say that the negative divisors of 15 are -1, -3, -5, and –15.

#### Prime Divisors of 15

The divisors of 15 are 1, 3, 5, and 15. Among these numbers, only 3 and 5 are prime numbers. So we obtain that:

The prime divisors of 15 are 3 and 5.

Key Things

The prime factorization of 15 is given below.

15 = 31×51

(i) By the number of divisors formula, we have that the number of divisors of 15 is

=(1+1)(1+1)=2×2=4.

(ii) By the sum of divisors formula, we have that the sum of the divisors of 15 is

$=\frac{3^2-1}{3-1} \times \frac{5^2-1}{5-1}$

$=\frac{9-1}{2} \times \frac{25-1}{4}$

$=4 \times 6=24$

(iii) By the product of divisors formula, we have that the product of the divisors of 15 is

=15(Number of divisors of 15)/3

=154/2

=152

#### Problem Solution on Divisors of 15

Question 1: What is the set of all divisors of 15 in roster form.

Solution:

As the numbers 1, 3, 5, and 15 only can divide 15, we conclude that the set of all divisors of the number 15 in roster form is

{1, 3, 5, 15}

Question 2: Write down the set of all prime divisors of 15 in roster form.

Solution:

We know that the prime numbers 3 and 5 only divide the number 15. Thus, the set of all prime divisors of 15 in roster form is as follows: {3, 5}.