Factors of 108 Using Prime Factorization
What are the factors of 108?
The integers that can divide a number evenly are known as factors of that number. When 108 is divided by its factor, the remainder is zero, and the quotient is also a factor of 108.
Factor list of the number 108
108 is a composite number, that is, it has more than two factors. The factors of 108 can be obtained using divisibility rules and division facts.
Therefore, the factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.
The Prime factors of 108
The factor trees below depict the prime factors of the number 108.
Hence, the prime factorization of 108 = 2 \(\times\) 2 \(\times\) 3 \(\times\) 3 \(\times\) 3
= \(2^2 \times 3^3\)
Therefore, 2 and 3 are the prime factors of 108.
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The positive pair factors of 108
A factor pair of 108 is a set of two integers, such that their product is 108. A number can have a positive pair of factors and a negative pair of factors.
For example, (1, 108) and (- 1, – 108) are factor pairs of 108.
1 \(\times\) 108 = 108
(- 1) \(\times\) (- 108) = 108
[Note: The product of two negative integers is positive.]
Below is a list of positive factor pairs of 108.
So, the positive factor pairs of 108 are 1 and 108, 2 and 54, 3 and 36, 4 and 27, 6 and 18, 9 and 12.
Solved Factors of 108 Examples
1) Find the common factors of 108 and 36.
Solution:
The factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Hence, the common factors of 108 and 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
2) Find the common factors of 108 and 45.
Solution:
The factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.
The factors of 45 are 1, 3, 5, 9, 15, and 45.
Hence, the common factors of 108 and 45 are 1, 3, and 9.
3) Find the common factors of 108 and 35.
Solution:
The factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.
The factors of 35 are 1, 5, 7, and 35.
Hence, the common factor of 108 and 35 is only 1.
4) Find the common factors of 108 and 37.
Solution:
The factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.
The factors of 37 are 1 and 37.
Hence, the common factor of 108 and 37 is only 1.
1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108 are the factors of 108.
2 × 2 × 3 × 3 × 3 or \(2^2\times3^3\) is the prime factorization of 108.
(1, 108), (2, 54), (3, 36), (4, 27), (6, 18) and (9, 12) are the positive pair factors of 108.
(- 1, – 108), (- 2, – 54), (- 3, – 36), (- 4, – 27), (- 6, – 18) and (- 9, – 12) are the negative pair factors of 108.
Yes, 54 is a factor of 108. 54 divides 108 exactly and leaves the remainder 0, hence 54 is a factor of 108.
The number 108 has a total of twelve factors: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.