Factors of 23 Using Prime Factorization
What is meant by Factors?
A factor of a number is an integer that divides the number evenly, that is, with no remainder. In other words, we can say that if a number is divided by its factor, the remainder is zero. The quotient obtained in this division operation is also a factor of the number.
What are the Factors of 23?
The factors of 23 are those integers that divide 23 without leaving any remainder. In other words, we can say that the factors of 23 will divide 23 exactly. Additionally, factors cannot be decimals or fractions. The quotient obtained on dividing 23 by its factor is also a factor of 23.
The factors of 23
Factors | Pair Factor | Prime factorization |
|---|---|---|
1, 23 | (1, 23) | 1 x 23 = 23 |
For example, 23 is a factor of 23 because when 23 is divided by 23, we get the remainder as 0 and 1 as the quotient. 1 is also a factor of 23.
Factor List for 23
We usually apply divisibility rules as well as division facts to create a list of factors for a number. In this case we can observe that divisibility rules do not apply to most natural numbers within 10 for the number 23.
Divisor | Is the number a factor of 23? | Multiplication Equation |
|---|---|---|
1 | Yes, 1 is a factor of every number | 1 x 23 = 23 |
2 | No, 11 is odd. | - |
3 | No, 1 + 1 = 2 is not divisible by 3 | |
4 | No, 23 \(\div\) 4 = 5R3 | - |
5 | No, the ones digit is not 0 or 5. | - |
6 | No, 23 is odd and not divisible by 3. | - |
7 | No, 23 \(\div\) 7 = 3 R2 | - |
8 | No, 23 \(\div\) 8 = 2 R7 | - |
9 | No, 23 \(\div\) 9 = 2 R5 | - |
Hence, we can stop checking the factor pairs after 9 as the multiplication equations will start to repeat.
Hence, the factors of 23 are 1 and 23. Therefore, 23 is a prime number as it has only 2 factors: 1 and itself.
The Prime Factorization for 23
A factor tree can be used to determine the prime factorization of 23, as shown in the image.
From the factor tree, we can see that the factorization of 23 is 1 x 23
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The Factor Pairs of 23
The factor pair of 23 is a set of two factors of 23 that when multiplied, give 23 as the product.
As 23 is a prime number, the only factor pair of 23 is,
Factors of 23 | Pair factor of 23 |
|---|---|
1, 23 | (1, 23) |
Solved Factors of 23 Examples
Example 1: Jennifer has a collection of 70 books. All the books are to be arranged on 23 shelves. Can she arrange all the books such that there are an equal number of books on each shelf with no books left behind? Explain your findings.
Solution:
70 books are to be evenly arranged on 23 shelves.
To find out the number of books that can be placed on each shelf, we divide 70 by 23,
\(\frac{70}{23}\) = 3 R1
On dividing 70 by 23, we get a non-zero remainder, which indicates that 23 is not a factor of 70. We get the quotient as 3 and the remainder as 1. Therefore, Jennifer can place 3 books each on a shelf, but she would be left with 1 book.
Hence, Jennifer cannot arrange all the books equally without any books left behind.
Example 2: Find the common factors of 29 and 23.
Solution:
Factors of 29 = 1, 29
Factors of 23 = 1, 23
So, the common factor of 23 and 29 is 1.
Hence, 29 and 23 have only one common factor.
Example 3: What are the common factors of 23 and 26?
Solution:
The factors of 23 = 1, 23
The factors of 26 = 1, 2, 13, 26
So, the common factor of 23 and 26 is 1.
Yes, 1 is a factor of 23. This is because the number 1 divides 23 exactly, that is, we get 0 as the remainder. So, 1 is a factor of 23.
No, the factors of 23 are 1 and 23. This suggests that it is a prime number. Therefore, 23 is not a composite number as it has exactly two factors, 1 and itself.
The factors of 23 are 1 and 23. So, the smallest factor of 23 is 1, and the greatest factor is 23 itself.
24, the factors of 23 are 1 and 23 and once added, these factors result in a sum of 24.
2, 23 is a prime number, so it has exactly 2 factors, 1 and 23.