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Factors of 108 are the integers that divide the number 108 completely. The numbers that can be split into 108 evenly are called factors of 108. There are 12 factors for 108 which are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108. The factors of 108 leave zero as the remainder on division with the number 108. The prime factorization of 108 is 2 × 2 × 3 × 3 × 3 = 22 × 33. 2 and 3 are the two prime factors of 108. When two numbers are multiplied together and they give 108 as the product, then, those numbers are called the pair factors of 108. The pair factors of 108 are (1, 108), (2, 54), (3, 36), (4, 27), (6, 18), and (9, 12).
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Key Terms: Factors of 108, Prime Factorization, Pair Factors, Remainder, Prime Factors, Prime Numbers, Composite Number
What are Factors of 108?
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Factors of 108 are the numbers that, on dividing 108, leave no remainder. They divide 108 exactly with zero as the remainder. 108 is a composite number which means it has many factors other than 1 and 108. 108 has a total of 12 factors. 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108 are the factors of 108.
Factors of 108
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Factors of 108 in Pairs
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Pair Factors of 108 are a pair of numbers that when multiplied together result in the original number 108. Pair factors of 108 can be positive as well as negative as the negative pair factors would also result in the positive value of 108 only. The positive and negative pair factors of 108 are as follows:
Positive Pair Factors of 108
The positive pair factors of 108 are:
| Positive Factors of 108 | Positive Pair Factors of 108 |
|---|---|
| 1×108 | (1, 108) |
| 2×54 | (2, 54) |
| 3×36 | (3, 36) |
| 4×27 | (4, 27) |
| 6×18 | (6, 18) |
| 9×12 | (9, 12) |
Read More: NCERT Solutions for Class 6 Maths Playing With Numbers
Negative Pair Factors of 108
The negative pair factors of 108 are:
| Negative Factors of 108 | Negative Pair Factors of 108 |
|---|---|
| -1×-108 | (-1, -108) |
| -2×-54 | (-2, -54) |
| -3×-36 | (-3, -36) |
| -4×-27 | (-4, -27) |
| -6×-18 | (-6, -18) |
| -9×-12 | (-9, -12) |
How to Calculate Factors of 108?
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Factors of 108 can be easily calculated by the basic division method. According to the division method, the factors of 108 are found by dividing 108 by different integers. If the given integers divide 108 exactly and leave zero as the remainder, then that integer is a factor of 108. Here is how we will find the factors of 108 through division method:
- 108/1 = 108 (Factor = 1; Remainder = 0)
- 108/2 = 54 (Factor = 2; Remainder = 0)
- 108/3 = 36 (Factor = 3; Remainder = 0)
- 108/4 = 27 (Factor = 4; Remainder = 0)
- 108/6 = 18 (Factor = 6; Remainder = 0)
- 108/9 = 12 (Factor = 9; Remainder = 0)
- 108/12 = 9 (Factor = 12; Remainder = 0)
- 108/18 = 6 (Factor = 18; Remainder = 0)
- 108/27 = 4 (Factor = 27; Remainder = 0)
- 108/36 = 3 (Factor = 36; Remainder = 0)
- 108/54 = 2 (Factor = 54; Remainder = 0)
- 108/108 = 1 (Factor = 108; Remainder = 0)
In case we divide 108 by any numbers other than 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108, then these numbers would leave a remainder. Thus, the only factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.
Read More: Greatest Common Divisor
Prime Factorization of 108
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Prime Factorization of 108 helps us to express the number 108 as the product of its prime factors. Here are the steps to find the prime factors of 108.
- Divide 108 by the smallest prime number which is 2. 108/2 = 52
- Now divide 52 again by 2. 52/2 = 27
- We can see that 27 is an odd number and cannot be divided by 2. Thus, divide 27 by the next prime number 3. 27/3 = 9
- Next, divide 9 by 3. 9/3 = 3
- Divide 3 by 3 at last. 3/3 =1
- We have received 1 at the end and we cannot proceed further, since 1 is not divisible by any prime number.
Prime Factorization of 108
Therefore, the prime factorization of 108 is 2 × 2 × 3 × 3 × 3 or 22 × 33. There are only two prime factors of 108 which are 2 and 3.
Read More: Co Prime Numbers
Factor Tree of 108
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Factor Tree is a special branching diagram where the factors of a number are expressed and the ends of a factor tree are all the prime factors of the original number.
The factor tree of 108 is as follows:
Factor Tree of 108
Solved Examples on Factors of 108
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Example 1: What are the common factors of 108 and 109?
Solution: The factors of 108 and 109 are:
- Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.
- Factors of 109: 1 and 109.
The common factor of 108 and 109 is 1 only.
Example 2: List the common factors of 108 and 54.
Solution: We will first write down all the factors of 108 and 54.
- Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.
- Factors of 54: 1, 2, 3, 6, 9, 18, 27, and 54.
Thus, the common factors of 108 and 54 are 1, 2, 3, 6, 9, 18, 27, and 54.
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| Related Topics | ||
|---|---|---|
| Multiplication and Division of Integers | Relation Between HCF and LCM | HCF |
| Properties of LCM and HCF | Perfect Numbers | Prime Number Formula |
Things to Remember
- Factors of 108 are the numbers that divide 108 completely leaving zero remainder.
- Factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.
- The factors of 108 in pairs are (1, 108), (2, 54), (3, 36), (4, 27), (6, 18), and (9, 12).
- The sum of the factors of 108 is 280.
- The prime factorization of 108 is 2 × 2 × 3 × 3 × 3.
- 2 and 3 are the two prime factors of the number 108.
Sample Questions
Ques. What are the common factors of 108 and 107? (2 Marks)
Ans. The factors of 108 and 107 are:
- Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.
- Factors of 107: 1, and 107.
Since 107 is a prime number, there is only one common factor of 108 and 107 which is 1.
Ques. Calculate the sum of all the factors of 108. (2 Marks)
Ans. We know that,
Factors of 108: 1, 2, 3 ,4, 6, 9, 12, 18, 27, 36, 54, 108
Adding up all of them, 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 27 + 36 + 54 + 108 = 280
Thus, the sum of all the factors of 108 is 280.
Ques. Calculate the mean of all the factors of 108. (2 Marks)
Ans. The Factors of 108 are 1, 2, 3 ,4, 6, 9, 12, 18, 27, 36, 54, and 108
To calculate the mean of all the factors of 108, we will divide the sum of the factors by the total number of factors.
Sum of the factors of 108: 1+ 2 + 3 + 4 + 6 + 9 + 12+ 18+ 27 + 36 + 54 + 108 = 280
280 ÷ 12 = 23.33
Thus, the mean of the factors of 108 is 23.3
Ques. What is the Least Common Multiple (LCM) and Greatest Common Factor (GCF) of 108 and 124? (2 Marks)
Ans. The factors of 108 and 124 are:
- Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
- Factors of 124: 1, 2, 4, 31, 62, 124.
Thus, the Least Common Multiple (LCM) of 108 and 124 is 3348 while the Greatest Common Factor (GCF) of 108 and 124 is 4.
Ques. What are the common factors of 108 and 25? (2 Marks)
Ans. We know that the factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108, and factors of 25 are 1, 5, and 25.
Thus, 108 and 25 have only one common factor which is 1 which means that they are coprime numbers.
Ques. Name the Greatest Common Factor of 108 and 50. (2 Marks)
Ans. The factors of 108 and 50 are
- Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
- Factors of 50: 1, 2, 5, 10, 25, 50
Thus, the common factors of 108 and 50 are 1, and 2. Thus, the Greatest Common Factor of 108 and 50 is 2.
Ques. What are the prime factors of 108? (1 Mark)
Ans. The prime factors of 108 are 2, and 3.
Ques. Write down the factors of 108. (1 Mark)
Ans. There are 12 factors of 108 which are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.
Ques. What are the positive and negative pair factors of 108? (2 Marks)
Ans. The positive pair factors of 108 are (1, 108), (2, 54), (3, 36), (4, 27), (6, 18) and (9, 12) while the negative pair factors of 108 are (-1, -108), (-2, -54), (-3, -36), (-4, -27), (-6, -18) and (-9, -12).
Ques. Check if 3 is a factor of 108. (2 Marks)
Ans. Yes, 3 is a factor of 108 as 3 divides 108 exactly and leaves the remainder 0.
108 / 3 = 36
Thus, 3 is a factor of 108.
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