Factors of 105

The factors of 105 are defined as the numbers which when divided to 105 should not leave any remainder. This means, 105 must be completely divisible by such a number that it does not leave any remainder.

• The factors of a number are finite. It means, universally, that number will have a particular number of factors only.
• Composite numbers can only be factored in because they are made from the product of two or more numbers.
• Only integers and whole numbers can be factored in and will have factors.
• Every factor of a number is equal to less than the number itself.

The integer 105 can be divided only into three numbers – 1, 3, 5, 7. It means, the product of 1, 3, 5, 7 will result in the number 105.

Moreover, the number 105 will have the following factors: 1, 3, 5, 7, 15, 21, 35 and 105. Therefore, the largest factor of 105 is 35, which is a composite number.

The largest prime factor of 105 is 7. The number 105 can be written as: 105 = 3 x 5 x 7. However, 105 will have the following categories of factors-

Positive factors: 1, 3, 5, 7, 15, 21, 35 and 105 Prime factors: 1, 3, 5, 7 Negative factors: -3, -5, -7 Prime factorization of 105 = 3 x 5 x 7

How to find the factors of 105?

The factors of 105 can be found by two methods –Long-division method and Prime factorization method.

Prime factorization method to find factors of 105

The prime factorization method is the most common method to represent several numbers, including 105.

Prime factorization also has another form, the prime factor tree method, which is a visual method to represent factors of 105.

It helps in determining various prime factors of 105. A factor tree is not unique for a given number. Instead of expressing 105 as 35 x 3, or 21 x 5, we can represent 105 as 3 x 5 x 7.

We need to divide 105, from the smallest number to the largest number so that it completely divides the number-

• With the help of divisibility rules, we need to find out the smallest factor of the given number. Here, we have 105, which is an odd number. It is not divisible by 2. In other words, we need to look for another number greater than 2 that can divide 105 completely.
• We can see the sum of 105 = 6. From the divisibility rule for number 3, we can find that it may be divisible by 3.
• 3 divides the number 105, therefore, it is the smallest prime factor of 105.
• 105 ÷ 3 = 35. We need to repeat the steps with other numbers until we get the remainder of 0.
• 35 can be divided into two more parts, i.e. 5 and 7.
• 35 ÷ 5 = 7, and 7 ÷ 7 = 1. Here, we get remainder 1. After this, our process is completed, and we get factors of 105.

Prime factorization of 105 using the prime factorization method is 3 x 5 x 7.

Example 1:

What are the common factors of 35 and 105?

Solution:

The factors of 35 are 7 and 5. The factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105. Thus, the common factors of 35 and 105 are 7 and 5.