Factors of 128 Using Prime Factorization
The Factors of 128
What are the Factors of 128?
The natural numbers that divide 128 without leaving any remainder are known as the factors of 128.
For example, 4 is a factor of 128 because when we divide 128 by 4, it leaves 32 as the quotient and 0 as the remainder. Here, the quotient is also a factor of 128.
So, to check if the number is a factor of 128 or not, divide 128 by that number and verify whether the remainder is zero or not.
Factor List of 128
Divisibility rules and division facts can be used to determine the factors of any number.
We can stop after checking for 9 because the factor pairs start to repeat.
Hence, the factors of 128 are 1, 2, 4, 8, 16, 32, 64, and 128.
The Prime Factors of 128
The number 128 is a composite number, that is, it has more than two numbers as factors. To find the prime factors, first we will divide the number 128 by its smallest prime factor, that is, 2.
128 ÷ 2 = 64
Again, divide by 2:
64 ÷ 2 = 32
32 ÷ 2 = 16
16 ÷ 2 = 8
8 ÷ 2 = 4
4 ÷ 2 = 2
2 ÷ 2 = 1
So, the prime factorization of 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 or \(2^7\).
This means that 2 is the prime factor of 128.
The Factor Pairs of 128
The factor pair of a number is a set of two of its factors that, when multiplied together, result in the original number.
Example: (8, 16) is the factor pair of 128, as 8 x 16 = 128.
The factor pair can be a positive pair or a negative pair. The factor pairs of any number can be obtained from the list of factors of that number.
The Positive Factor Pairs of 128
Hence, the positive factor pairs of 128 are: (1, 128), (2, 64), (4, 32), and (8, 16).
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Rapid Recall
The factors of 128 are:
Solved Factors of 128 Examples
Example 1: Find the common factors of 128 and 120.
Solution:
The factors of 128 = 1, 2, 4, 8, 16, 32, 64, and 128
The factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120
Therefore, the common factors of 128 and 120 are 1, 2, 4, and 8.
Example 2: Find the common factors of 128 and 131.
Solution:
The factors of 128 = 1, 2, 4, 8, 16, 32, 64, and 128
The factors of 131 = 1 and 131
Therefore, the common factor of 128 and 131 is 1.
Example 3: Find the greatest common factor of 8 and 128.
Solution:
Factors of 8 = 1, 2, 4, and 8
Factors of 128 = 1, 2, 4, 8, 16, 32, 64, and 128
Therefore, the greatest common factor of 8 and 128 is 8.
Example 4: Joanna wants to plant 128 saplings with her mom. She wants to decide the best way to divide the saplings into rows and columns, to make them look attractive. In how many ways can Joanna arrange the saplings in the garden?
Solution:
The factor pairs of 128 would give us ways to arrange or divide the saplings into rows and columns.
The factor pairs of 128 are:
1 x 128 = 128
2 x 64 = 128
4 x 32 = 128
8 x 16 = 128
So the factor pairs of 128 are (1, 128), (2, 64), (4, 32), and (8, 16), that is, a total of 4 factor pairs.
Each factor pair provides 2 ways to arrange the saplings.
Therefore, Joanna can arrange the saplings in 4 x 2 = 8 ways.
The factors of 128 are 1, 2, 4, 8, 16, 32, 64, and 128
So, the smallest factor of 128 is 1, and the greatest factor is 128 itself.
The factors of 128 are 1, 2, 4, 8, 16, 32, 64, and 128
Sum of factors = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255
So, the sum of the factors of 128 is 255.
Yes, 16 is a factor of 128. This is because when the number 16 divides 128, it leaves 8 as the quotient and 0 as the remainder.
Prime numbers are the numbers that have only two factors: 1 and the number itself.
The factors of 128 are 1, 2, 4, 8, 16, 32, 64, and 128.
Here, 128 has more than two factors.
Therefore, 128 is not a prime number
No, 128 \(\div\) 7 = 18R2. As we divide the number 128 by 7, it leaves 18 as the quotient and 2 as the remainder. So, 7 is not a factor of 128.
A composite number is a natural number with more than two factors.
For example, the factors of 8 are 1, 2, 4, and 8
Since the number 8 has more than two factors, it is a composite number.