# Divisors of 72

The divisors of 72 are those numbers that completely divide 72 without a remainder. In this section, we will discuss about divisors of 72.

#### Highlights of Divisors of 72

• Divisors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
• Negative divisors of 72:1, -2, -3, -4, -6, -8, -9, -12, -18, -24, -36 and -72.
• Prime divisors of 72: 2 and 3
• Number of divisors of 72: 12
• Sum of divisors of 72: 195
• Product of divisors of 72: 72

#### What are Divisors of 72

A number n is a divisor of 72 if $\frac{72}{n}$ is an integer. Note that if 72/n=m is an integer, then both m and n will be the divisors of 72.

To find the divisors of 72, we need to find the numbers n such that 72/n becomes an integer. We have:

 72/1=72 1, 72 are divisors of 72. 72/2=36 2, 36 are divisors of 72. 72/3=24 3, 24 are divisors of 72 72/4=18 4, 18 are divisors of 72 72/6=12 6, 12 are divisors of 72 72/8=9 8, 9 are divisors of 72

No numbers other than 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72 can divide 72. So we conclude that

 The divisors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.

Thus, the total number of divisors of 72 is twelve.

Divisors of 60: The divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

Divisors of 64: The divisors of 60 are 1, 2, 4, 8, 16, 32, 64.

Divisors of 75: The divisors of 75 are 1, 3, 5, 15, 25, 75.

Divisors of 100: The divisors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100.

#### Negative Divisors of 72

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

As the divisors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72, we can say that:

The negative divisors of 72 are -1, -2, -3, -4, -6, -8, -9, -12, -18, -24, -36 and -72.

#### Prime Divisors of 72

The divisors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72. Among these numbers, only 2 and 3 are prime numbers. So we obtain that:

The prime divisors of 72 are 2 and 3.

Key Things

The prime factorization of 72 is given below.

72 = 23 × 32

(i) By the number of divisors formula, we have that the number of divisors of 72 is

=(3+1)(2+1)=4×3=12.

(ii) By the sum of divisors formula, we have that the sum of the divisors of 72 is

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

$=\frac{16-1}{1} \times \frac{27-1}{2}$

$=15 \times 13=195$

(iii) By the product of divisors formula, we have that the product of the divisors of 72 is

=72(Number of divisors of 72)/2

=7212/2

=726

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