Factors of 51 Using Prime Factorization
What are the Factors of 51?
If the number 51 is divided by a natural number such that the remainder is zero, the natural number is known as a factor of 51. Here, the quotient obtained on division is also a factor of 51.
The factors of 51 are 1, 3, 17, and 51 because all these numbers divide the number 51 evenly.
List of the Factors of 51
Divisibility rules and division facts can be used to determine the factors of 51.
Divisor | Is the number a factor of 51? | Multiplication equation |
|---|---|---|
1 | Yes, 1 is a factor of every number. | 15 \(\times \) 1 = 51 |
2 | No, 51 is not even. | - |
3 | Yes, 5 + 1 = 6 is divisible by 3. | 3 \(\times \) 17 = 51 |
4 | No, 51 \(\div \) 4 = 12 R3 | - |
5 | No, ones digit is neither 0 nor 5. | - |
6 | No, 51 is divisible by 3 but not an even number. | - |
7 | No, 51 \(\div \) 7 = 7 R2 | - |
8 | No, 51 \(\div \) 8 = 6 R3 | - |
9 | No, 51 \(\div \) 9 = 5 R6 | - |
We can stop the derivation of factors with this step because the multiplication equations will start to repeat. So, the factors of 51 are the multiplicands of each multiplication equation. Since there are 2 multiplication equations with unique multiplicands, the number 51 has four factors, which are, 1, 3, 17 and 51.
[Note: If we divide 51 by 20, we get the quotient and remainder as 2 and 11, respectively. So, 20 is not a factor of 51. Similarly, you can check for other numbers as well.]
The Prime Factorization of 51
If a number can be expressed as a product of prime numbers, these prime numbers are known as the prime factors of that number. The process of writing this multiplication of prime factors is known as prime factorization.
The prime factorization of 51 can be represented by using a factor tree as shown below.
So, the prime factorization of 51 is 51 = 3 \(\times \) 17, and the prime factors of 51 are 3 and 17.
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Factor Pairs of 51
A factor pair of a number is a set of any of its two factors such that their product is the number itself. A factor pair can be negative or positive.
Positive Factor Pairs of 51
Here we can observe the factor pairs of 51. There are two factor pairs for 51.
Positive Factors of 51 | Positive Factor Pairs of 51 |
|---|---|
1 \(\times \) 51 | (1, 51) |
3 \(\times \) 17 | (3, 17) |
Solved Factors of 51 Examples
Example 1: Find the common factors of 50 and 51
Solution:
Factors of 50 = 1, 2, 5, 10, 25, and 50
Factors of 51 = 1, 3, 17, and 51
Therefore, the common factor of 50 and 51 is 1.
Example 2: Find the greatest common factor of 45 and 51.
Solution:
Factors of 45 = 1, 3, 5, 9, 15, and 45
Factors of 51 = 1, 3, 17, and 51.
So, the common factors of 45 and 51 are 1 and 3.
Therefore, the greatest common factor of 45 and 51 is 3.
Example 3: Harry was studying geometric shapes and found a rectangle of area 51 square centimeters. What are the possible dimensions of the rectangle?
Solution:
The area of the rectangle is \(51~cm^2\).
Area of rectangle = length x width
51 = length x width
So, the dimensions of the rectangle can be obtained from factor pairs of 51, which are, (1, 51) and (3, 17).
Each factor pair can be arranged in 2 ways.
Therefore, the possible dimensions of the rectangle are:
Length in centimeters | Width in centimeters |
|---|---|
1 | 51 |
3 | 17 |
17 | 3 |
51 | 1 |
The factor pairs of 51 are the pairs of numbers whose product is 51. The factor pairs of 51 are (1, 51) and (3, 17).
Composite numbers are those numbers which have a factor other than 1 and themselves, that is, a composite number has more than two factors.
The division of any number by 1 results in zero as the remainder, and the quotient is the number itself. Hence, 1 is a factor of all numbers.
The prime factors of 51 are 3 and 17.
Therefore the sum of prime factors of 51 is 3 + 17 = 20.