What Are the Factors of 1 and Why Are They Important?
The integers that divide a given number evenly and without leaving a remainder are known as its factors. Every number is its own factor. 1 is a factor of every number and every number has at least 2 factors: 1 and the number itself. The factor of 1 is 1.
What is the Factor of 1?
The integers that divide the given number 1 evenly and without leaving a remainder are known as its factors. 1 has a unique factor which is 1 itself. There are negative factors as well but generally whenever we talk about factors we consider only positive numbers.
The given topic factor of 1 is taken from the section factors from 1 to 100.
Factors of 1 in Pair
Factor pair of a number is a combination of two numbers such that the product of the given number gives the original number.
The factors of 1 in pairs are pairs of numbers that when multiplied together result in 1. Thus, the factor of 1 in pairs is (1, 1). The above pair of numbers when multiplied together results in 1. There is also the negative factor of 1 that can also be paired. The negative factor of 1 in pairs is (-1, -1).
Factor of Prime Number From 1 to 100
Prime number is a number that is divisible by 1 and itself. So, the positive factors of a prime number are 1 and itself. We know that the negative value of factors of a number is also a factor of that number.
The prime numbers from 1 to 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. All of the above numbers have two positive factors and two negative factors.
Prime Factorization of 1
Every number can be expressed as a product of its prime factors according to the fundamental rule of arithmetic; this process is known as the prime factorization of a number.
The factor of the given number 1 is 1 itself. It did not have any prime factorization. As a result, it is also possible to write 1 as the product of the factors 1 and 1. The factor tree of 1 can be used to perform the prime factorization of 1.
Tree Diagram of 1
The factor tree can be represented in a variety of ways, but the most popular ones are listed here as the factor tree of 1.
Common Factors of 1 and Any Number
Any number is divisible by 1 without leaving any remainder. Thus, 1 is a factor of any number. We know that the negative value of a factor is also a factor of that number. Thus, 1 and -1 are factors of any number. Again, 1 and -1 are only two factors of 1. Thus, the common factors of 1 and any number are 1 and -1. The factors of 1 are 1 and -1.
Let's take an example: Find common factors of 1 and 30.
The positive factors of 30 are 1, 2, 3, 5, 10, 15, 30.
Factors of 30 are 1, 2, 3, 5, 10, 15, 30,-1, -2, -3, -5, -10, -15,- 30.
Therefore, 1 and -1 are the common factors of 1 and 30.
Take some more examples: Find common factors of 1 and 5.
The positive factors of 5 are 5 and 1.
All factors of 5 are 1, 5, -1, -5.
The common factors of 1 and 5 are 1 and -1, which are factors of 1.
Key Features of the Factor of 1
The factor of a number is an exact divisor of that number. It does not give any remainder. The factor of 1 is 1 which is an exact divisor of 1.
The factor of 1 neither gives prime factor nor conjugate factors.
Every number can be expressed as a product of its prime factors according to the fundamental rule of arithmetic. The factor of 1 does not have any prime factorization.
Solved Examples
1. What is the sum of factors of 1?
Solution: The factors of 1 are 1 and -1.
The sum of factors of 1 is 1 + (-1) = 0.
2. Find the factors of 5.
Solution: 5 is a prime number. The factor of 5 is 1 and 5. Also, the negative value of the factors is also a factor of the number.
The factors of 5 are 1, 5, -1, and -5.
3. What is the common factor of 1 and 2?
Solution: The factors of 1 are 1, -1. The factors of 2 are 1, 2, -1, and -2.
The common factors of 1 and 2 are 1 and -1.
Practice Problems
1. Find the prime factors of 12.
Answer: 2, 3
2. Find the common factors of 13 and 15.
Answer: 1 and -1.
Conclusion
The factor of 1 is 1. The prime factorization of 1 gives only 1 because the factor of 1 is 1 itself. 1 is the factor of all the numbers. The common factor of 1 and any number always gives only 1 as a common factor.
FAQs on Factors of 1: Complete Guide for Students
1. What are the factors of 1?
The number 1 has only one factor: 1 itself. A factor is a whole number that divides another number exactly, without leaving a remainder. Since only the number 1 can divide 1 perfectly, it is its only factor.
2. How do we find the factors of 1?
To find the factors of 1, you need to identify all the whole numbers that divide 1 without a remainder. The only whole number that satisfies this condition is 1. You can also think of it in terms of multiplication: the only pair of whole numbers that multiplies to give 1 is 1 × 1. Therefore, the only factor is 1.
3. Why is the number 1 considered neither prime nor composite?
The number 1 is unique and is classified as neither prime nor composite based on the standard definitions used in mathematics. Here's the explanation:
- A prime number must have exactly two distinct positive factors: 1 and the number itself. For example, 7 has factors 1 and 7.
- A composite number must have more than two positive factors. For example, 4 has factors 1, 2, and 4.
Since the number 1 has only one positive factor (itself), it does not meet the criteria for being prime or composite.
4. What is unique about the number 1 in terms of its factors?
The most unique property of the number 1 is that it is the only positive integer that has exactly one factor. Every other positive integer has at least two factors (1 and itself). This special status is also linked to its role as the multiplicative identity, meaning any number multiplied by 1 remains unchanged.
5. What are the positive and negative factors of 1?
While students usually focus on positive factors, integers can also have negative factors. For the number 1:
- The only positive factor is 1.
- The only negative factor is -1, because (-1) × (-1) = 1.
Therefore, the complete set of integer factors for 1 is {-1, 1}.
6. What is the prime factorization of 1?
The number 1 does not have a prime factorization. Prime factorization is the process of expressing a number as a product of its prime factors. Since 1 is not a prime number and cannot be formed by multiplying any prime numbers together, it is considered an exception and has no prime factors.
7. How are factors different from multiples?
Factors and multiples are related but opposite concepts.
- Factors are the numbers that divide a given number exactly. The factors of 8 are 1, 2, 4, and 8. A number has a finite set of factors.
- Multiples are the numbers you get when you multiply a given number by other integers. The multiples of 8 are 8, 16, 24, 32, and so on. A number has an infinite set of multiples.
For the number 1, its only factor is 1, but its multiples are all the integers (1, 2, 3, ...).
