Factors of 216 Using Prime Factorization
What are the Factors of 216?
The natural numbers that can divide a number evenly are known as the factors of that number. When 216 is divided by its factor, the remainder is zero, and the quotient is also a factor of 216.
The number 216 has a total of sixteen factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216.
Factor List of the Number 216
The factors of 216 can be obtained using divisibility rules and division facts.
Number | Is the number a factor of 216? | Multiplication equation |
|---|---|---|
1 | Yes, 1 is a factor of every number | 1 x 216 = 216 |
2 | Yes, 216 is even | 2 x 108 = 216 |
3 | Yes, 2 + 1 + 6 = 9 is divisible by 3 | 3 x 72 = 216 |
4 | Yes, 216 \(\div\) 4 = 54 R 0 | 4 x 54 = 216 |
5 | No, 216 \(\div\) 5 = 43 R 2 | - |
6 | Yes, 216 is even and divisible by 3 | 6 x 36 = 216 |
7 | No, 216 \(\div\) 7 = 30 R 6 | - |
8 | Yes, 216 \(\div\) 8 = 27 R 0 | 8 x 27 = 216 |
9 | Yes, 2 + 1 + 6 = 9 is divisible by 9 | 9 x 24 = 216 |
10 | No, 216 \(\div\) 10 = 21 R 2 | - |
11 | No, 216 \(\div\) 11 = 19 R 12 | - |
12 | Yes, 216 \(\div\) 12 = 18 R 0 | 12 x 18 = 216 |
Therefore, the factors of 216 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216.
The Prime Factors of a Number
The factor tree below depicts the prime factorization of the number 216.
The prime factorization of 216 is 2 × 2 × 2 × 3 × 3 × 3 or \(2^3\) × \(3^3\). This means 2 and 3 are the prime factors of 216.
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The Positive Factor Pairs of 216
A factor pair of 216 is a set of two factors of 216 such that their product is 216. A number can have a positive pair of factors and a negative pair of factors.
For example, (1, 216) is a factor pair of 216.
1 x 216 = 216
A list of positive factor pairs of 216 is given below:
So, the factor pairs for 216 are 1 and 216, 2 and 108, 3 and 72, 4 and 54, 6 and 36, 8 and 27, 9 and 24, 12 and 18.
Solved Factors of 216 Examples
Example 1: Find the common factors of 216 and 42.
Solution:
The factors of 216 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216.
The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.
Hence, the common factors of 216 and 42 are 1, 2, 3, and 6.
Example 2: Find the greatest common factor of 216 and 432.
Solution:
The factors of 216 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216.
The factors of 432 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, 216, and 432.
So the common factors of 216 and 432 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216.
Hence, the greatest common factor of 216 and 432 is 216.
Example 3: Find the total number of common factors of 216 and 210.
Solution:
The factors of 216 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216.
The factors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, and 210.
So the common factors of 216 and 210 are 1, 2, 3, and 6.
Hence, the total number of common factors of 216 and 210 is 4.
Example 4: Tom buys 216 wall stickers. He wants to paste the stickers on a wall in a rectangular array. How many different arrays can he make?
Solution:
To find the number of arrays Tom can make, find the number of factor pairs of 216.
The number of factor pairs of 216 = 8
Each factor pair can make 2 arrays.
So, the number of different arrays = The number of factor pairs x 2
= 8 x 2
= 16 [Multiply]
Hence, Tom can paste the stickers into 16 different rectangular arrays.
The number 216 has 16 factors, that is, it has more than two factors. Hence, 216 is a composite number.
2 × 2 × 2 × 3 × 3 × 3 or \(2^3\)× \(3^3\) is the prime factorization of 216.
The number 216 has more than two factors, so it is not a prime number.
(-1, -216), (-2, -108), (-3, -72), (-4, -54), (-6, -36), (-8, -27), (-9, -24), and (-12, -18) are the negative factor pairs of 216.
72 divides 216 exactly and leaves the remainder 0. Hence, 72 is a factor of 216.