How to Find All Factors of 58 with Step-by-Step Solutions
The numbers that divide 58 evenly without producing a remainder are known as the factors of 58. It is possible to represent the factors and pair factors of 58 in both positive and negative forms but not in decimal or fractional forms. The pair factors of 58 are shown as (1, 58) or (-1, -58) and (2,29) or (-2,-29). When two negative numbers are multiplied together, for example, -1 and -58, the result is 58.
This article will cover the factors of 58, the positive and negative pair factors of 58, and how to use the prime factorization method to discover the prime factors of 58.
What are the Factors of 58?
The factors of 58 are the integers that divide 58 evenly and have a remainder of 0. In other words, the factors of 58 are the integers that, when multiplied in pairs, resulting in the original number 58. 58 is a composite number that is even, therefore, it has more than two factors. Hence, 1, 2, 29, and 58 are the factors of 58. Similarly, -1, -2, -29, and -58 are the negative components of 58.
Factors of 58 are 1, 2, 29 and 58.
The negative value of the factors is also a factor of 58.
So, -1,-2,-29, and -58 are also factors of 58.
The factors of 58 are 1,2,29,58,-1,-2,-29,-58.
Prime Factorization of 58
The number 58 is represented as the product of its prime components when divided into its prime factors. Find the prime factors of 58 by following the steps below. Consider a pair factor of 58, for example (1, 58). The number 58, in this instance, is an even composite number that may be further divided into its prime factors. First, we will divide 58 by 2. Then we will try to divide the quotient by a number.
\[58\div 2= 29\]
29 is not divisible by 2.
Similarly, 29 is not divisible by 3,4,5,6,...,28.
Hence the prime factorization of 58 is $2 \times 29.$
Prime Factors of 58
Prime numbers are those with just two factors, namely 1 and the number itself. The factors of 58 are 1, 2, 29, and 58. The prime numbers among these are 2 and 29. In this case, 2 and 29 are 58's prime factors.
Pair Factors of 58
The pair factors of 58 are the two numbers that, when multiplied together, equal the number 58. The pair factors of 58 that are positive and negative are as follows.
Hence, the positive pairs of 58 are (1, 58) and (2,29). The negative pairs of 58 are (-1, -58) and (-2, -29).
Tree Diagram
The tree diagram of 58
Factors of 58 by Division Method
The division method can be used to find all the factors of 58. The number 58 can be divided in this way by several integers. The integers are the factors of 58 if they divide 58 evenly and without leaving a residue. Let's discuss how to divide 58 to discover its factors using the division method.
\[58\div 1 = 58\] (Remainder is 0)
\[58\div 2 = 29\] (Remainder is 0)
\[58\div 29 = 2\] (Remainder is 0)
\[58\div 58 = 1\] (Remainder is 0)
Try to divide 58 with any other integers. You will not receive a quotient with a remainder of 0. Hence through this method, the factors of 58 are found as 1, 2, 29, and 58.
Solved Examples
1: What is the HCF of 58 and 70?
Ans: Initially find the prime factors of 58 and 70.
Prime factors of 58 are 2, 29. Prime factorization of 58 done below:
\[58 = 2 \times 29\]
Prime factors of 70 are 2, 5, 7. Prime factorization of 70 is done below:
\[70 = 2\times 5 \times 7\]
The HCF is the highest common factor of the two numbers. Here, the highest common factor between 58 and 70 is 2. Hence the HCF of 58 and 70 is 2.
2: Find the common factors of 58 and 31.
Ans: Here, the factors of 58 are 1, 2, 29 and 58.
The factors of 31 are 31 and 1.
Hence, the common factor of 58 and 31 is 1.
3: What is the sum of all positive factors of 58?
Ans: Here, the factors of 58 are 1, 2, 29 and 58.
Sum = 1 + 2 + 29 + 58 = 90
Hence the sum of all factors of 58 is 90.
4: Find the common factors of 58 and 100.
Ans: Finding the factors of 58:
\[58 = 1\times 2\times29\]
The prime factors of 58 are 2,29
\[100 = 1\times 2 \times 2\times 5\times 5\]
The prime factors of 100 are 2 and 5.
The factors of 100 are \[1, 2, 2\times 2, 5, 2\times 5, 2\times 2 \times5 , 5\times 5, 2\times 5\times 5, 5\times 5 \times 2 \times 2\].
The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.
Hence, the common factor of 58 and 100 are 1 and 2.
Conclusion
In this topic page, we get knowledge of factors of the number 58 and also the prime factor of the number 58. By using a tree diagram, we can get factors and prime factors of 58.
Key Features
To find the factors of 58 we can use a factor tree.
The factors of 58 are 1, 2, 29, 58 and their negative counterparts.
The factor pairs of 58 are \[1 \times 58\], \[2\times 29\], \[-1\times -58\] and \[2\times-29\].
Practice Problems
1. What is the sum of all factors of 29?
Answer: 0
2. What are the factors of 20?
Answer: 1, 2, 4, 5, 10, 20, -1, -2, -4, -5, -10, -20.
List of Related Articles
FAQs on What Are the Factors of 58?
1. What are all the factors of 58?
The factors of 58 are integers that can divide 58 without leaving a remainder. The factors include both positive and negative numbers. The positive factors are 1, 2, 29, and 58. The complete list of factors is -58, -29, -2, -1, 1, 2, 29, and 58.
2. How can you find the factors of 58 using the division method?
To find the factors of 58 using the division method, you divide 58 by integers starting from 1 up to 58 to see which divisions result in a zero remainder.
- 58 ÷ 1 = 58 (Remainder 0), so 1 and 58 are factors.
- 58 ÷ 2 = 29 (Remainder 0), so 2 and 29 are factors.
3. What are the prime factors of 58?
The prime factors of a number are the prime numbers that divide it exactly. To find the prime factors of 58, we can use prime factorization. Since 58 is an even number, we start by dividing by the smallest prime number, 2.
58 = 2 × 29
Both 2 and 29 are prime numbers. Therefore, the prime factors of 58 are 2 and 29.
4. How can the factors of 58 be written as pairs?
Factor pairs are pairs of integers that multiply together to give the original number. The factor pairs for 58 are:
- Positive Pairs: (1, 58) and (2, 29)
- Negative Pairs: (-1, -58) and (-2, -29)
5. Is 58 a prime or a composite number, and why?
A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. Since the factors of 58 are 1, 2, 29, and 58, it has four positive factors. Because it has more than two factors, 58 is a composite number.
6. Why are negative numbers also considered factors of 58?
The mathematical definition of a factor is any integer that divides another integer perfectly, leaving no remainder. This definition includes negative numbers. For example, 58 can be divided by -2 to get -29, with a remainder of zero. Similarly, the product of two negative factors, like (-2) × (-29), equals positive 58. Therefore, negative integers that satisfy these conditions are also factors.
7. How can you quickly check if any number, for example, 4, is a factor of 58?
To check if a number is a factor of 58, you simply divide 58 by that number. If the division results in a whole number (i.e., the remainder is 0), then the number is a factor. For example, to check if 4 is a factor:
58 ÷ 4 = 14.5
Since the result is not a whole number (it has a remainder of 2), 4 is not a factor of 58.
8. What is the sum of all the positive factors of 58?
This is an application-based question often found in aptitude tests. First, list all the positive factors of 58, which are 1, 2, 29, and 58. Next, add them together:
1 + 2 + 29 + 58 = 90
Therefore, the sum of all positive factors of 58 is 90.
