# Divisors of 54

The divisors of 54 are those numbers that completely divide 54 with the remainder zero. In this section, we will discuss about divisors of 54.

#### Highlights of Divisors of 54

• Divisors of 54: 1, 2, 3, 6, 9, 18, 27 and 54
• Negative divisors of 54: -1, -2, -3, -6, -9, -18, -27 and -54
• Prime divisors of 54: 2 and 3
• Number of divisors of 54: 8
• Sum of divisors of 54: 120
• Product of divisors of 54: 544

#### What are Divisors of 54

A number n is a divisor of 54 if $\frac{54}{n}$ is an integer. Note that if 54/n=m is an integer, then both m and n will be the divisors of 54.

To find the divisors of 54, we need to find the numbers n such that 54/n becomes an integer. We have:

 54/1=54 1, 54 are divisors of 54. 54/2=27 2, 27 are divisors of 54 54/3=18 3, 18 are divisors of 54 54/6=9 6, 9 are divisors of 54

No numbers other than 1, 2, 3, 6, 9, 18, 27 and 54 can divide 54. So we conclude that

 The divisors of 54 are: 1, 2, 3, 6, 9, 18, 27 and 54.

Thus, the total number of divisors of 54 is eight.

#### Negative Divisors of 54

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

As the divisors of 54 are 1, 2, 3, 6, 9, 18, 27 and 54, we can say that:

The negative divisors of 54 are -1, -2, -3, -6, -9, -18, -27 and -54.

#### Prime Divisors of 54

The divisors of 54 are 1, 2, 3, 6, 9, 18, 27 and 54. Among these numbers, only 2 and 3 are prime numbers. So we obtain that:

The prime divisors of 54 are 2 and 3.

Key Things

The prime factorization of 54 is given below.

54 = 21 × 33

(i) By the number of divisors formula, we have that the number of divisors of 54 is

=(1+1)(3+1)=2×4=8.

(ii) By the sum of divisors formula, we have that the sum of the divisors of 54 is

$=\frac{2^2-1}{2-1} \times \frac{3^4-1}{3-1}$

$=\frac{4-1}{1} \times \frac{81-1}{2}$

$=3 \times 40=120$

(iii) By the product of divisors formula, we have that the product of the divisors of 54 is

=54(Number of divisors of 54)/2

=548/2

=544

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