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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