Factors of 216: Prime Factorization, Methods, Tree, and Examples

The factors of 216 are the numbers that are exactly divisible by 216 without any remainder. The number 216 is an even composite number, also not a perfect square.
Factors of two sixteen

Figure 1 – All possible Factors of 216

This article explores the various methods used in finding the factors of 216. 216 is a composite number, implying that it will have multiple factors other than 1 and 216.  This makes the task challenging, if not tricky, but once you have reached the end of this article, you will have mastered all the different techniques mentioned in it.

What Are the Factors of 216?

The entire factors of 216 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216. Factors are the numbers that output a zero remainder when divided by 216. The number 216 has a total of 16 factors, and these 16 factors can be paired up into 8-factor pairs. 

How To Calculate the Factors of 216?

The factors of 216 can be calculated using multiple methods that are both mathematical and visual, such as the division method and prime factorization method. We will look for a number that divides 216 ultimately with zero remainders; this number becomes a factor of 216. 

Division

While all these methods are simple, the most traditional is the division method. This method is based on seeking a perfect divisor for the given number once we find the number that yields zero after division with 216, both this number and the quotient are considered to be the factors of 216.  The factors of 216 through the division method are shown below: \[ \frac{216}{1} = 216\] \[ \frac{216}{2} = 208\] \[ \frac{216}{3} = 72\] \[ \frac{216}{4} = 54 \] \[\frac{216}{6} = 36 \] \[ \frac{216}{8} = 27 \] \[ \frac{216}{9} = 24\] \[ \frac{216}{12} = 18 \] And similarly, their respective whole number quotients also act as factors. Factors of 216: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 208, and 216. The second method is prime factorization, in which we perform division but only through prime numbers seeking a result equal to 1. The visual tool is a factor tree drawn of the prime factors of 216.

Factors of 216 by Prime Factorization

The second method of prime factorization is a technique that divides the number into the minor possible prime numbers. To understand this method, let us first brush up on our concept of prime numbers. Prime numbers are the numbers that are only divisible by one and the number itself. They do not have multiple factors like composite numbers; therefore, they only have two factors in total.  In prime factorization, we perform division only using prime numbers; if a number is not divisible by, let’s say, 5, we will not use six next for division. Instead, we will shift to the following prime number, 7. Once these factors are obtained, they are called prime factors.  When using the division method for finding factors of 216, we noticed that the factors were both composite and prime. But in prime factorization, we get only prime factors. The equation of prime factorization of 216 can be written as follows:

Prime Factorization of 216 =  2 x 2 x 2 x 3 x 3 x 3

Prime Factorization of 216 = $2^{3}$ x $3^{3}$

A detailed explanation of prime factorization, and we reached this above equation, has been provided in example 1 in the last section of this article. The prime factorization of 216 is also shown below:
Prime factorization of two hundred and sixteen

Figure 2 – Prime Factorization of 216

Factor Tree of 216

A factor tree is a visual tool for understanding and representing the prime factors of a number. Using a visual aid, this method expresses the factors of any number, such as 216, in the shape of a tree. It is just another method for finding out the factors of a number through the branching of a tree. This tree keeps branching into factors until no more factorization is possible. Factorization further becomes impossible once we reach a prime number.  It is better if one has familiarized oneself with the prime factors of a number before drawing the factor tree. We start by writing the original number at the head of the tree, which is then divided into two branches; at the end of these branches, we report the resultant two factors after the first division. The steps continue onwards till all prime numbers are reached. In this section, we will also look at the factor tree of 216; we can observe from figure 2 below that the branches start extending out from 216 and stop at prime numbers as shown:
Factor tree of two hundred and sixteen

Figure 3 – Factor Tree of 216

Factors of 216 in Pairs

Pair Factors are a pair of factors of any number, as the name indicates, that gives the original number when multiplied together. The factor pair of 216 is explained as follows: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216 Considering the condition for a factor pair, we know that (2, 108) will make a factor pair because 2 x 108 = 216. Similarly,

(3, 72) make a factor pair because 3 x 72 = 216

(4, 54) make a factor pair because 4 x 54 = 216

(6, 36) make a factor pair because 6 x 36 = 216

(8, 27) make a factor pair because 8 x 27 = 216

(9, 24) make a factor pair because 9 x 24 = 216

(12, 18) make a factor pair because 12 x 18 = 216

So, the factor pairs of 216 are: Factor pairs of 216 = (1, 216), (2, 108), (3, 72), (4, 54), (6, 36), (8, 27), (9, 24), (12, 18) Every number has both positive and negative factor pairs, so the negative factor pairs of 216 are: Negative Factor pairs of 216 = (-1, -216), (-2, -108), (-3, -72), (-4, -54), (-6, -36), (-8, -27), (-9, -24), (-12, -18) From the factor pairs mentioned above, we can infer that the only difference between the positive and negative factor pairs is the sign. While writing positive factor pairs, we ignore the (+) sign with all the integers, but for negative factor pairs, it is necessary that we include the (-) sign with all the factors.

Factors of 216 Solved Examples

Let us now attempt to further solidify our learning by applying it to examples. In these examples, we will use methods like prime factorization and division to find out the factors of 216.

Example 1

Use the factorization method to find out the entire prime factors of 216.

Solution

In the factorization method, also known as the prime factorization method, we find out the prime factors of any number, such as 216, by dividing with prime numbers. Following is the process of finding these prime factors:

Step 1: $\frac{216 }{ 2} = 108$ (108 is an even number hence divisible by 2)

Step 2: $\frac{108}{ 2} = 54$ (54 is an even number hence divisible by 2)

Step 3: $\frac{54}{  2} = 27$ (27 is not an even number, so we try the next prime number after 2 which is 3)

Step 4: $\frac{27}{ 3} = 9$ (9 is an odd number and divisible by 3 as well)

Step 5: $\frac{9}{3} = 3$ (3 is an odd number divisible by the prime number 3)

Step 6: $\frac{3}{ 3} = 1$ (Our factorization is complete)

Once we reach the answer of 1, the factorization method is complete. To note down the prime factors of 216, we will look at the divisors at each step, and these divisors are 2, 2, 2, 3, 3, 3. These are also known as the prime factors of 216. The prime factors of 216 through factorization are 2, 2, 2, 3, 3, 3. The prime factorization equation can also be written as:

Prime factorization equation = 2 x 2 x 2 x 3 x 3 x 3

Example 2

Calculate the average of all the factors of 216. 

Solution

Factors of 216 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216 216 has a total of 16 factors. Now, as we have the factors of 216, let us calculate the average of these factors. The formula used for calculating the average of factors of 216 is:

Average of factors of 216 = Sum of all factors / Total number of factors

Sum of all factors of 216 = 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 27 + 36 + 54+72+108+216

Sum of all factors of 216 = 600

Average of factors = 600 / total number of factors

Total number of factors = 16

Average of factors = 600 / 16

Average of factors of 216 = 37.5

Hence, the average of factors of 216 is 37.5. All images/mathematical drawings are created with GeoGebra. 

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