Factors of 420 Using Prime Factorization
What are the Factors of 420?
If the number 420 is divided by a natural number such that the remainder is zero, then the natural number is known as a factor of 420. Here, the quotient obtained on division is also a factor of 420.
The factors of 420 are 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210 and 420 because all these numbers divide the number 420 evenly.
List of the Factors of 420
By applying both the divisibility rule for different natural numbers and division facts, we can list the factors of 420.
Divisor | Is the number a factor of 420? | Multiplication equation |
|---|---|---|
1 | Yes, 1 is a factor of every number | 1 x 420 = 420 |
2 | Yes, 420 is even | 2 x 210 = 420 |
3 | Yes, 4 + 2 + 0 = 6 is divisible by 3 | 3 x 140 = 420 |
4 | Yes, as 420 \(\div\) 4 = 105 R0 | 4 x 105 = 420 |
5 | Yes, ones digit is 0 | 5 x 84 = 420 |
6 | Yes, 420 is even and divisible by 3 | 6 x 70 = 420 |
7 | Yes, as 420 \(\div\) 7 = 60 R0 | 7 x 60 = 420 |
8 | No, as 420 \(\div\) 8 = 52 R4 | - |
9 | No, as 420 \(\div\) 9 = 46 R6 | - |
10 | Yes, ones digit is 0 | 10 x 42 = 420 |
11 | No, as 420 \(\div\) 11 = 38 R2 | - |
12 | Yes, 420 \(\div\) 12 = 35 R0 | 12 x 35 = 420 |
13 | No, 420 \(\div\) 13 = 32 R4 | - |
14 | Yes, 420 \(\div\) 14 = 30 R0 | 14 x 30 = 420 |
15 | Yes, 420 \(\div\) 15 = 28 R0 | 15 x 28 = 420 |
16 | No, 420 \(\div\) 16 = 26 R4 | - |
17 | No, 420 \(\div\) 17 = 24 R12 | - |
18 | No, 420 \(\div\) 18 = 23 R6 | - |
19 | Yes, 420 \(\div\) 19 = 22 R2 | - |
20 | Yes, 420 \(\div\) 20 = 21 R0 | 20 x 21 = 420 |
21 | No, 420 \(\div\) 21 = 20 R0 | 21 x 20 = 420 |
The multiplication equation has repeated at 21. Hence, we can stop checking the factors after 21 as we have derived the factor list of 420.
So, the factors of 420 are 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210 and 420
Prime Factorization of 420
If a natural number can be expressed as a product of prime numbers, these prime numbers are known as the prime factors of that natural number. The process of expressing a number as the product of its prime factors is known as prime factorization.
The prime factorization of 420 can be represented by using a factor tree as shown below:
So, the prime factorization of 420 is 420 = 2 x 2 x 3 x 5 x 7.
Read More:
Factor pairs of 420
A factor pair of a number is a set of two factors of a number such that their product is the number itself.
Positive Factor Pairs of 420
Positive Factors of 420 | Positive Factor Pairs of 420 |
|---|---|
1 x 420 | (1, 420) |
2 x 210 | (2, 210) |
3 x 140 | (3, 140) |
4 x 105 | (4, 105) |
5 x 84 | (5, 84) |
6 x 70 | (6, 70) |
7 x 60 | (7, 60) |
10 x 42 | (10, 42) |
12 x 35 | (12, 35) |
14 x 30 | (14, 30) |
15 x 28 | (15, 28) |
20 x 21 | (20, 21) |
Rapid Recall
The factors of 420 are 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210 and 420.
Solved Factors of 420 Examples
Example 1: Find the greatest common factor of 420 and 150.
Solution:
Factors of 420 = 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210 and 420.
Factors of 150 = 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, and 150.
Therefore, the greatest common factor of 420 and 150 is 30.
Example 2: Find the common factors of 420 and 300.
Solution:
Factors of 420 = 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210 and 420.
Factors of 300 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, and 300.
Therefore, the common factors of 420 and 300 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 30 and 60.
Example 3: What is the sum of the factors of 420?
Solution:
Factors of 420 = 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210 and 420.
Therefore, sum of factors of 420 are 1 + 2 + 3 + 4 + 5 + 6 + 7 + 10 + 12 + 14 + 15 + 20 + 21 + 28 + 30 + 35 + 42 + 60 + 70 + 84 + 105 + 140 + 210 + 420 = 1344.
Example 4: 420 customers can be seated in a restaurant. But, only 2 people can be accommodated on each table. For Valentine’s day, the restaurant offered a special discount on desserts, where each table receives an extra dessert for free on ordering one. It is known that all the tables ordered one dessert, how many desserts were served?
Solution:
Total number of customers in the restaurant = 420.
Number of tables in the restaurant = \(\frac{420}{2}\) = 210.
Each table orders a dessert. Due to the offer, they get an extra dessert. Hence each table gets two desserts.
Total number of desserts = 210 x 2 = 420.
Therefore, the number of desserts served is 420.
A number that has only two factors, that is, 1 and itself is called a prime number. For example, 2, 3, 7, 19 and so on.
Composite numbers are numbers that have more than two factors. For example, 4, the factors of 4 are 1, 2 and 4.
The division of any positive number by 1 results in zero as the remainder, and the quotient is the number itself. Hence, 1 is a factor of all positive numbers.
A number which divides the given number exactly, that is leaving a remainder of 0, is called a factor of the number.