Factors of 105

Factors of 105 are numbers that divide it without leaving any remainder. These factors can be positive or negative. Pair factors of 105 are combinations of factors that can also be positive or negative, like 1 and 105, or -1 and -105. Multiplying negative pair factors, such as -1 and -105, gives us the original number 105. In this article, we will explore the factors, pair factors, and prime factors of 105 using the prime factorization method, with several examples for clarity.

Introduction to Factors of 105

Factors of 105 are the numbers that divide 105 without any remainder. These factors can be either positive or negative. For example, the pair factors of 105 can be 1 and 105, or -1 and -105. Multiplying negative factors, such as -1 and -105, will still result in 105.

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Factors of 105

Factors of 105 are the numbers that divide 105 exactly, leaving no remainder. In other words, factors of 105 are pairs of numbers that, when multiplied, give 105. Since 105 is a composite number, it has multiple factors besides one and itself. The positive factors of 105 are 1, 3, 5, 7, 15, 21, 35, and 105. The negative factors are -1, -3, -5, -7, -15, -21, -35, and -105.

Factors of 105: 1, 3, 5, 7, 15, 21, 35, and 105
Prime Factorization of 105: 3, 5, 7

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Pair Factors of 105

Pair factors of 105 are two numbers that multiply to give the number 105. These pair factors can be positive or negative. Here are the positive and negative pair factors of 105:

Positive Pair Factors of 105

Negative Pair Factors of 105

Factor Tree of 105

A factor tree is a method used to find a number's prime factorization. In factor trees, we first identify the factors of a number and then further break down those factors until we reach prime numbers. The first step in creating a factor tree is to find a pair of factors whose product equals the number we are factoring. These two factors form the first branch of the factor tree. We generally choose different pairs of factors to start the process. We then repeat the process with each factor until all branches of the tree end in prime numbers.

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Prime Factorization of 105

The number 105 can be expressed as the product of its prime factors. Here's how to find the prime factors of 105:

• Assume a pair factor of 105 is (1, 105).
• Factor 1 is neither prime nor composite and cannot be split further.
• The other factor, 105, is a composite number that can be broken down into prime factors.
• 105 can be written as the product of 3 and 35.
• Three is a prime number, and thirty-five is a composite number. Divide 35 into its prime factors.
• 35 can be written as the product of 5 and 7, both of which are prime numbers.
• Now, write all the prime factors together.

So, 105 is written as 3, 5, and 7.

Thus, the prime factorization of 105 is 3, 5, and 7.

Factors of 105 by Division Method

The division method finds the factors of 105 by dividing it by different integers. If an integer divides 105 exactly, with a remainder of 0, then it is a factor of 105. Let's start dividing 105 by different integers.

105 ÷ 1 = 105 (Factor is 1, Remainder is 0)
105 ÷ 3 = 35 (Factor is 3, Remainder is 0)
105 ÷ 5 = 21 (Factor is 5, Remainder is 0)
105 ÷ 7 = 15 (Factor is 7, Remainder is 0)
105 ÷ 15 = 7 (Factor is 15, Remainder is 0)
105 ÷ 21 = 5 (Factor is 21, Remainder is 0)
105 ÷ 35 = 3 (Factor is 35, Remainder is 0)
105 ÷ 105 = 1 (Factor is 105, Remainder is 0)