The divisors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. The divisors of 72 are those numbers that completely divide 72 without a remainder. In this section, we will discuss divisors of 72.
Table of Contents
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: 726
What are Divisors of 72
A number n is a divisor of 72 if $\dfrac{72}{n}$ 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 the following:
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.
Also Read:
| 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 64 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.
Video Solution of Divisors of 72:
Sum, Product & Number of Divisors of 72
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
$=\dfrac{2^4-1}{2-1} \times \dfrac{3^3-1}{3-1}$
$=\dfrac{16-1}{1} \times \dfrac{27-1}{2}$
= 15 × 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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FAQs
Answer: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 are all divisors of 72.
Answer: Note that 72 = 23 × 32. So the sum of divisors of 72 equals (24-1)/(2-1) ×(33-1)/(3-1) = 15 × 13 = 195.
