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Factors of 378: Prime Factorization, Methods, and Examples
In this article, we are going to discuss factors of the number 378.
Factors of 378 are positive as well as negative numbers such that when these factors divided by 378 give no residue in this particular we have 16 positive factors and 16 negative factors ,in total there are 32 factors for number 378.
Factors of 378
Here are the factors of number 378.
Factors of 378: 1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, 189, and 378
Negative Factors of 378
The negative factors of 378 are similar to their positive aspects, just with a negative sign.
Negative Factors of 378: -1, -2, -3, -6, -7, -9, -14, -18, -21, -27, -42, -54, -63, -126, -189, and -378
Prime Factorization of 378
The prime factorization of 378 is the way of expressing its prime factors in the product form.
Prime Factorization: 2 x 3 x 3 x 3 x 7
In this article, we will learn about the factors of 378 and how to find them using various techniques such as upside-down division, prime factorization, and factor tree.
What Are the Factors of 378?
The factors of 378 are 1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, 189, and 378. These numbers are the factors as they do not leave any remainder when divided by 378.
The factors of 378 are classified as prime numbers and composite numbers. The prime factors of the number 378 can be determined using the prime factorization technique.
How To Find the Factors of 378?
You can find the factors of 378 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 378, create a list containing the numbers that are exactly divisible by 378 with zero remainders. One important thing to note is that 1 and 378 are the 378’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 378 are determined as follows:
\[\dfrac{378}{1} = 378\]
\[\dfrac{378}{2} = 189\]
\[\dfrac{378}{3} = 126\]
\[\dfrac{378}{6} = 63\]
\[\dfrac{378}{7} = 54\]
\[\dfrac{378}{9} = 42\]
\[\dfrac{378}{14} = 27\]
\[\dfrac{378}{18} = 21\]
\[\dfrac{378}{21} = 18\]
\[\dfrac{378}{27} = 14\]
\[\dfrac{378}{42} = 9\]
\[\dfrac{378}{54} = 7\]
\[\dfrac{378}{63} = 6\]
\[\dfrac{378}{126} = 3\]
\[\dfrac{378}{189} = 2\]
\[\dfrac{378}{378} = 1\]
Therefore, 1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, 189, and 378 are the factors of 378.
Total Number of Factors of 378
For 378, there are 16 positive factors and 16 negative ones. So in total, there are 32 factors of 378.
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 378 is given as:
Factorization of 378 is 2 x 3 x 3 x 3 x 7.
The exponent of 2, 3, 3, 3, and 7 is 1.
Adding 1 to each and multiplying them together results in 32.
Therefore, the total number of factors of 378 is 32. 16 is positive, and 16 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 378 by Prime Factorization
The number 378 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 378 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 378, 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 378 can be expressed as:
378 = 2 x 3 x 3 x 3 x 7
Factors of 378 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 378, the factor pairs can be found as:
1 x 378 = 378
2 x 189 = 378
3 x 126 = 378
6 x 63 = 378
7 x 54 = 378
9 x 42 = 378
14 x 27 = 378
18 x 21 = 378
The possible factor pairs of 378 are given as (1, 378),(2, 189), (3, 126),(6, 63), (7, 54),(9, 42), (14, 27),and (18, 21).
All these numbers in pairs, when multiplied, give 378 as the product.
The negative factor pairs of 378 are given as:
-1 x -378 = 378
-2 x -189 = 378
-3 x -126 = 378
-6 x -63 = 378
-7 x -54 = 378
-9 x -42 = 378
-14 x -27 = 378
-18 x -21 = 378
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, -7, -9, -14, -18, -21, -27, -42, -54, -63, -126, -189, and -378 are called negative factors of 378.
The list of all the factors of 378, including positive as well as negative numbers, is given below.
Factor list of 378: 1, -1, 2, -2, 3, -3, 6, -6, 7, -7, 9, -9, 14, -14, 18, -18, 21, -21, 27, -27, 42, -42, 54, -54, 63, -63, 126, -126, 189, -189, 378, and -378.
Factors of 378 Solved Examples
To better understand the concept of factors, let’s solve some examples.
Example 1
How many factors of 378 are there?
Solution
The total number of Factors of 378 is 16.
Factors of 378 are 1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, 189, and 378.
Example 2
Find the factors of 378 using prime factorization.
Solution
The prime factorization of 378 is given as:
378 $\div$ 2 = 189
189 $\div$ 3 = 63
63 $\div$ 3 = 21
21 $\div$ 3 = 7
7 $\div$ 7 = 1
So the prime factorization of 378 can be written as:
2 x 3 x 3 x 3 x 7 = 378