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Factors of 217: Prime Factorization, Methods, and Examples
The factors of 217 are the numbers that when divided by 217 leave nothing as the remainder. Also, 217 is an odd composite number. It has 4 positive factors in total.
The factors of the given number can be positive as well as negative provided that the given number is achieved upon multiplication of two-factor integers.
Factors of 217
Here are the factors of number 217.
Factors of 217: 1, 7, 31, 217
Negative Factors of 217
The negative factors of 217 are similar to its positive factors, just with a negative sign.
Negative Factors of 217: -1, -7, -31, -217
Prime Factorization of 217
The prime factorization of 217 is the way of expressing its prime factors in the product form.
Prime Factorization: 7 x 31
In this article, we will learn about the factors of 217 and how to find them using various techniques such as upside-down division, prime factorization, and factor tree.
What Are the Factors of 217?
The factors of 217 are 1, 7, 31, and 217. All of these numbers are the factors as they do not leave any remainder when divided by 217.
The factors of 217 are classified as prime numbers and composite numbers. The prime factors of the number 217 can be determined using the technique of prime factorization.
How To Find the Factors of 217?
You can find the factors of 217 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 217, create a list containing the numbers that are exactly divisible by 217 with zero remainders. One important thing to note is that 1 and 217 are the 217’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 217 are determined as follows:
\[\dfrac{217}{1} = 217\]
\[\dfrac{217}{7} = 31\]
\[\dfrac{217}{31} = 7\]
\[\dfrac{217}{217} = 1\]
Therefore, 1, 7, 31, and 217 are the factors of 217.
Total Number of Factors of 217
For 217 there are 4 positive factors and 4 negative ones. So in total, there are 8 factors of 217.
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 217 is given as:
Factorization of 217 is 1 x 7 x 31.
The exponent of 1, 7, and 31 is 1.
Adding 1 to each and multiplying them together results in m.
Therefore, the total number of factors of 217 is 8. 4 are positive and 4 factors are negative.
Important Notes
Here are some important 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 217 by Prime Factorization
The number 217 is a composite. Prime factorization is a useful technique for finding the number’s prime factors and expressing the number as the product of its prime factors.
Before finding the factors of 217 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 217, start dividing by its smallest 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 217 can be expressed as:
217 = 7 x 31
Factors of 217 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 of the given numbers.
For 217, the factor pairs can be found as:
1 x 217 = 217
7 x 31 = 217
The possible factor pairs of 217 are given as (1, 217) and (7, 31 ).
All these numbers in pairs, when multiplied, give 217 as the product.
The negative factor pairs of 217 are given as:
-1 x -217 = 217
-7 x -31 = 217
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, -7, -31, and -217 are called negative factors of 217.
The list of all the factors of 217 including positive as well as negative numbers is given below.
Factor list of 217: 1, -1, 7, -7, 31, -31, 217, and -217
Factors of 217 Solved Examples
To better understand the concept of factors, let’s solve some examples.
Example 1
How many factors of 217 are there?
Solution
The total number of Factors of 217 is 4.
Factors of 217 are 1, 7, 31, and 217.
Example 2
Find the factors of 217 using prime factorization.
Solution
The prime factorization of 217 is given as:
217 $\div$ 7 = 31
31 $\div$ 31 = 1
So the prime factorization of 217 can be written as:
7 x 31 = 217