Factors of 105 Using Prime Factorization
What are the Factors of 105?
The integers that can divide a number exactly are known as the factors of that number. When a number is divided by its factor the remainder is zero and the quotient is also a factor of that number.
The number 105 has a total of eight factors: 1, 3, 5, 7, 15, 21, 35 and 105.
Factor List of 105
105 is a composite number and so it has more than two factors.
Therefore, the factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105.
Prime Factorization of 105
We see that the factor tree can be used to find the prime factors of 105.
Prime factorization of 105 is \(3 \times 5 \times 7\). This shows us that 3, 5 and 7 are the prime factors of 105.
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Positive Pair Factors of 105
A pair factor or factor pair of 105 is a set of two factors of 105 that when multiplied give the product as 105. The following are the positive pair factors of 105:
Solved Examples
Example 1: Find the common factors of 105 and 84.
Solution:
The factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105.
The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84.
Hence, the common factors of 105 and 84 are 1, 3, 7 and 21.
Example 2: Find the greatest common factor between 105 and 56.
Solution:
The factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105.
The factors of 56 are 1, 2, 4, 7, 8, 14, 28 and 56.
Hence, the common factors of 105 and 56 are 1 and 7. So the greatest common factor between 105 and 56 is 7.
Example 3: There are 15 boxes of candies. Each box contains 7 pieces of candy. These candies are to be distributed equally among 5 children. How many pieces of candy do they each get?
Solution:
Number of boxes of candy = 15
Number of pieces of candy in each box = 7
So, the total number of candies = 15 \(\times\) 7 = 105
Number of children = 5
Number of pieces of candy each child gets = \(\frac{105}{5}\) = 21
Hence, each child gets 21 pieces of candy.
1, 3, 5, 7, 15, 21, 35 and 105 are the factors of 105.
3 × 5 × 7 is the prime factorization of 105.
(1, 105), (3, 35), (5, 21), and (7, 15) are the positive pair factors of 105.
Yes, 21 is a factor of 105. 21 divides 105 exactly and leaves the remainder as 0, hence 21 is a factor of 105.