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Factors of 297: Prime Factorization, Methods, and Examples
The factors of 297 are the group of numbers that are divided by 297 and leave nothing as remainders. The number 297 is an odd number. It is also a composite number with eight factors in total.
Let us explore more about the factors of 297.
Factors of 297
Here are the factors of number 297.
Factors of 297: 1, 3, 9, 11, 27, 33, 99, and 297
Negative Factors of 297
The negative factors of 297 are similar to its positive aspects, just with a negative sign.
Negative Factors of 297: –1, -3, -9, -11, -27, -33, -99, and -297
Prime Factorization of 297
The prime factorization of 297 is the way of expressing its prime factors in the product form.
Prime Factorization: 3 x 3 x 3 x 11
In this article, we will learn about the factors of 297 and how to find them using various techniques such as upside-down division, prime factorization, and factor tree.
What Are the Factors of 297?
The factors of 297 are 1, 3, 9, 11, 27, 33, 99, and 297. These numbers are the factors as they do not leave any remainder when divided by 298.
The factors of 297 are classified as prime numbers and composite numbers. The prime factors of the number 297 can be determined using the prime factorization technique.
How To Find the Factors of 297?
You can find the factors of 297 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 297, create a list containing the numbers that are exactly divisible by 297 with zero remainders. One important thing to note is that 1 and 297 are 297’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 297 are determined as follows:
\[\dfrac{297}{1} = 297\]
\[\dfrac{297}{3} = 99\]
\[\dfrac{297}{9} = 33\]
\[\dfrac{297}{11} = 27\]
\[\dfrac{297}{297} = 1\]
Therefore, 1, 3, 9, 11, 27, 33, 99, and 297 are the factors of 297.
Total Number of Factors of 297
For 297, there are eight positive factors and eight negative ones. So in total, there are sixteen factors of 297.
To find the total number of factors of the given number, follow the procedure mentioned below:
- Find the factorization/prime 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 X is given as:
Factorization of 297 is 1 x 3$^3$ x 11.
The exponent of 1 and 11 is 1. The exponent of 3 is 3.
Adding 1 to each and multiplying them together results in 16.
Therefore, the total number of factors of 297 is sixteen. Eight are positive, and eight 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 297 by Prime Factorization
The number 297 is a composite. 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 297 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 297, 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 297 can be expressed as:
297 = 3 x 3 x 3 x 11
Factors of 297 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 297, the factor pairs can be found as:
1 x 297 = 297
3 x 99 = 297
9 x 33 = 297
11 x 27 = 297
The possible factor pairs of 297 are given as (1, 297), (3, 99), (9, 33), and (11, 27).
All these numbers in pairs, when multiplied, give 297 as the product.
The negative factor pairs of 297 are given as:
1 x 297 = 297
3 x 99 = 297
9 x 33 = 297
11 x 27 = 297
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, -3, -9, -11, -27, -33, -99, and -297 are called negative factors of 297.
Factors of 297 Solved Examples
To better understand the concept of factors, let’s solve some examples.
Example 1
How many factors of 297 are there?
Solution
The total number of Factors of 297 is eight.
Factors of 297 are 1, 3, 9, 11, 27, 33, 99, and 297.
Example 2
Find the factors of 297 using prime factorization.
Solution
The prime factorization of 297 is given as:
297 $\div$ 3 = 99
99 $\div$ 3 = 33
33 $\div$ 3 = 11
11 $\div$ 11 = 1
So the prime factorization of 297 can be written as:
3 x 3 x 3 x 11 = 297