Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

Factors of 1

ffImage
Last updated date: 17th Apr 2024
Total views: 153k
Views today: 4.53k
hightlight icon
highlight icon
highlight icon
share icon
copy icon

Factors of 1: An Introduction

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


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

1. Is the factor of 1 prime or not?

If a number is divisible by 1 and itself, then the number is a prime number. 1 is divisible by 1 and itself. According to modern Mathematics, a prime number must be divisible by exactly 2 numbers. Thus, 1 is not a prime number.

2. Find common positive factors of 11 and 21.

The positive factors of 11 are 1, 11.

The positive factors of 21 are 1, 21.

The common factor of 11 and 21 is 1.

We can directly conclude that both are prime numbers, so only 1 is a common factor.

3. Find the common factors of two prime numbers.

The factors of a prime number are 1, -1, the positive and negative values of the prime number. Assume that a and b are two prime numbers. Then, the factors of a are 1, -1, a, -a. The factors of b are 1, -1, b, and -b. So, 1 and -1 are common factors of a and b. So, the common factors of any two prime numbers are  1 and -1.