The Factors of 87 - Introduction
When we discuss the factor of 87, we truly mean that all the positive and negative integers can divide the number 87 with the remainder zero. The result of taking the number 87 and dividing it by one of its factors would be another factor. These numbers can be written as factor pairs. We present them both ways. This is a specific number's mathematical decomposition.
Let's examine how to identify and list each of the 87 elements.
Prime Factors of 87
Sometimes the prime factors of 87 and the prime factorisation of 87 are used interchangeably, which is strictly incorrect. Read this section carefully to understand the difference between the prime factors of 87 and the prime factorisation of 87.
3 x 29 is the prime factorisation of 87. This is a simple and unique list of prime factors of 87, together with their multiplicities. \[87 = 3 \times 29\].
The prime factors of 87 are the prime numbers that divide 87 exactly, with the remainder or modulo 0, in terms of Euclidean division.
3, 29 are the prime factors of 87.
The prime factorisation does not include 1, but it does include every copy of every prime factor.
Factor Pairs of 87
The various combinations of two numbers you multiply together to get 87 are known as the factor pairs of 87.
We know that any number is 1 time of itself.
So, the first pair of 87 is (1,87).
Next, we move to the 2. There is no number whose 2 times is 87.
Next move to 3. 3 times 29 is 87.
The next factor pair of 87 is (3,29).
After 3, we did not get two numbers whose product is 87.
Again,
\[-1 \times (-87)=87\]
\[-3 \times (-29) = 87\]
So, the factor pairs of 87 are (1,87),(3,29),(-1,-87) and (-3,-29).
What is a Factor Tree?
A factor tree is a graphic representation of the possible factors of a number and its sub-factors.
It is intended to make factorisation simpler. It is made by first determining the factors of a number and then the factors of the original number.
Once you get a large number of prime factors, which is the prime factorisation of the original number, the process recursively continues. Make sure to keep in mind the second component in a factor pair when building the tree. See below the factor tree of 87:
Factor tree of 87
The Least Common Factor of 87 with Another number
By comparing the prime factorization of the two numbers and taking the highest common prime factor, it is possible to find the biggest common factor between two numbers. If there is not a common factor, the HCF is always 1. This is one of the common prime factors of two numbers and is also known as the highest common factor. It is the biggest prime factor that the two numbers have in common. Any pair of integers has 1 as the least common factor (the smallest number they have in common).
Taking a number 2.
We can rewrite \[3 = 3 \times 1\]
Also, \[87= 1\times 3 \times 29\]
So the least common factor of 3 and 87 is 1.
Find the Number of Factor of 87
If a number N can be written as \[N = X^{a}Y^{b}Z^{c}\] where X, Y and Z are prime factors of N, then the number of factors is (a+1)(b+1)(c+1).
The prime factorization form of 87 is \[87 = 3^{1}\cdot 29^{1}\].
Thus, the number of positive factors of 87 is (1+1)(1+1) = 4.
Solved Examples
Example 1: Find the common factors of 87 and 232.
Sol:
\[87 = 3 \times 29\]
The factors of 87 are 1, 3, 29, and 87.
\[232 = 1 \times 2 \times 2 \times 2 \times 29\]
1, 2, 4, 8, 29, 58, 116, and 232 are the factors of 232.
Therefore, the common factors of 87 and 232 are 1 and 29.
Example 2: What is the sum of all the prime factors of 87.
Sol:
The prime factors of 87 are 3 and 29. So, the sum of all the prime factors of 87 are 3 + 29 = 32.
Example 3: Find the sum of all positive factors of 87.
Sol: The factors of 87 are 1,3,9, and 87. The sum of all positive factors of 87 is 1+3+29+87 = 120.
Key Features
87 is a composite number rather than a prime number.
The composite number 87 has the following prime factors: 87 = 3 × 29.
The factors 1, 3, 29, 87 and the negatives -1, -3, -29, -87 all these can exactly divide 87.
List of Related Articles
Practice on Your Own
Q 1: What are the positive factors of 87?
Ans: 1, 3, 29 and 87.
Q 2: What is the product of the prime factors of 87?
Ans: 87.
Q 3: Are 87 and 89 prime numbers?
Ans: 87 is not a prime number but 89 is a prime number.
FAQs on Factors of 87
1: What are all the factor pairs of 87?
List of factor pairs for 87
\[1 \times 87 = 87\]
\[3 \times 29 = 87\]
\[-29 \times -3 = 87\]
\[-87 \times -1 = 87\]
2: 87 is a prime number, isn't it?
No, 87 is not a prime number. There are divisors of 87: 1, 3, 29, 87. A number needs exactly two factors to be considered a prime number. 87 is not a prime number because it has more than two factors, including 1, 3, 29, and 87.
3: Why is it important to learn about the prime factorization of numbers?
Prime factorization is used to find two numbers' greatest common divisor (GCD).
The HCF is necessary when simplifying (reducing) fractions to the lowest terms.
Prime factorization is useful when calculating the least common multiple, LCM, of numbers; this is necessary when adding or subtracting ordinary fractions.