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Factors of 306: Prime Factorization, Methods, and Examples
The factors of 306 are the 12 numbers that leave a zero remainder when they act as the divisors for the number 306. They also yield a whole number quotient.
The given number’s factors can be positive and negative, provided that the given number is achieved upon multiplication of two-factor integers.
Factors of 306
Here are the factors of number 306.
Factors of 306: 1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, and 306
Negative Factors of 306
The negative factors of 306 are similar to its positive aspects, just with a negative sign.
Negative Factors of 306: -1, -2, -3, -6, -9, -17, -18, -34, -51, -102, -153, and -306
Prime Factorization of 306
The prime factorization of 306 is the way of expressing its prime factors in the product form.
Prime Factorization: 2 x $3^{2}$ x 17
In this article, we will learn about the factors of 306 and how to find them using various techniques such as upside-down division, prime factorization, and factor tree.
What Are the Factors of 306?
The factors of 306 are 1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, and 306. These numbers are the factors as they do not leave any remainder when divided by 306.
The factors of 306 are classified as prime numbers and composite numbers. The prime factors of the number 306 can be determined using the prime factorization technique.
How To Find the Factors of 306?
You can find the factors of 306 by using the rules of divisibility. The divisibility rule states that any number, when divided by any other natural number, is said to be divisible by the number if the quotient is the whole number and the resulting remainder is zero.
To find the factors of 306, create a list containing the numbers that are exactly divisible by 306 with zero remainders. One important thing to note is that 1 and 306 are the 306’s factors as every natural number has 1 and the number itself as its factor.
1 is also called the universal factor of every number. The factors of 306 are determined as follows:
\[\dfrac{306}{1} = 306\]
\[\dfrac{306}{2} = 153\]
\[\dfrac{306}{3} = 102\]
\[\dfrac{306}{6} = 51\]
\[\dfrac{306}{9} = 34\]
\[\dfrac{306}{17} = 18\]
Therefore, 1, 2, 3, 6, 9, and 17 and all their whole number quotients 18, 34, 51, 102, 153, and 306 are the factors of 306.
Total Number of Factors of 306
For 306, there are 12 positive factors and 12 negative ones. So in total, there are 24 factors of 306.
To find the total number of factors of the given number, follow the procedure mentioned below:
- Find the factorization of the given number.
- Demonstrate the prime factorization of the number in the form of exponent form.
- Add 1 to each of the exponents of the prime factor.
- Now, multiply the resulting exponents together. This obtained product is equivalent to the total number of factors of the given number.
By following this procedure, the total number of factors of 306 is given as:
Factorization of 306 is 1 x 2 x $3^{2}$ x 17.
The exponent of 1, 2, and 17 is 1 and that of 3 is 2.
Adding 1 to each and multiplying them together results in 24.
Therefore, the total number of factors of 306 is 24. 12 are positive, and 12 factors are negative.
Important Notes
Here are some essential points that must be considered while finding the factors of any given number:
- The factor of any given number must be a whole number.
- The factors of the number cannot be in the form of decimals or fractions.
- Factors can be positive as well as negative.
- Negative factors are the additive inverse of the positive factors of a given number.
- The factor of a number cannot be greater than that number.
- Every even number has 2 as its prime factor, the smallest prime factor.
Factors of 306 by Prime Factorization
The number 306 is a composite number. Prime factorization is a valuable technique for finding the number’s prime factors and expressing the number as the product of its prime factors.
Before finding the factors of 306 using prime factorization, let us find out what prime factors are. Prime factors are the factors of any given number that are only divisible by 1 and themselves.
To start the prime factorization of 306, start dividing by its most minor prime factor. First, determine that the given number is either even or odd. If it is an even number, then 2 will be the smallest prime factor.
Continue splitting the quotient obtained until 1 is received as the quotient. The prime factorization of 306 can be expressed as:
306 = 2 x $3^{2}$ x 17
Factors of 306 in Pairs
The factor pairs are the duplet of numbers that, when multiplied together, result in the factorized number. Factor pairs can be more than one depending on the total number of factors given.
For 306, the factor pairs can be found as:
1 x 306 = 306
2 x 153 = 306
3 x 102 = 306
6 x 51 = 306
9 x 34 = 306
17 x 18 = 306
The possible factor pairs of 306 are given as (1, 306), (2, 153), (3, 102), (6, 51), (9, 34), and (17, 18).
All these numbers in pairs, when multiplied, give 306 as the product.
The negative factor pairs of 306 are given as:
-1 x -306 = 306
-2 x -153 = 306
-3 x -102 = 306
-6 x -51 = 306
-9 x -34 = 306
-17 x -18 = 306
It is important to note that in negative factor pairs, the minus sign has been multiplied by the minus sign, due to which the resulting product is the original positive number. Therefore, -1, -2, -3, -6, -9, -17, -18, -34, -51, -102, -153, and -306 are called negative factors of 306.
The list of all the factors of 306, including positive as well as negative numbers, is given below.
Factor list of 306: 1, -1, 2, -2, 3, -3, 6, -6, 9, -9, 17, -17, 18, -18, 34, -34, 51, -51, 102, -102, 153, -153, 306 and -306
Factors of 306 Solved Examples
To better understand the concept of factors, let’s solve some examples.
Example 1
How many factors of 306 are there?
Solution
The total number of Factors of 306 is 12.
Factors of 306 are 1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, and 306.
Example 2
Find the factors of 306 using prime factorization.
Solution
The prime factorization of 306 is given as:
306 $\div$ 2 = 153
153 $\div$ 3 = 51
51 $\div$ 3 = 17
17 $\div$ 17 = 1
So the prime factorization of 306 can be written as:
2 x $3^{2}$ x 17 = 306