What Are Prime Factors and Why Are They Important in Maths?
Definition of A Factor
The word ‘Prime Factors’ is made up of two distinct terms, Prime and Factor, both of which are important in Mathematics. To understand the concept of prime factors properly, we need to understand what are factors in detail and then, move on to understand what are prime numbers. When we multiply two numbers, we get the product. Factors are numbers which have been multiplied to get the product. Likewise, factors can divide the product perfectly with no reminder.
Let us take a simple example:
If we take a look at number 16, then to get the number 16, we would have to multiply 8 with 2. So, in this case, 8 and 2 are factors of 16. On the other hand, if we divide 16 by 2, i.e., 16/2, we get the remainder of 0. So, a number when divided by any of its factors would always give us the remainder of 0 - that is, it will be perfectly divisible. One product can have multiple factors as well like 16 can also have factors 4,4 (4 x 4) and 16,1 (16 x 1).
What is a Prime Number
Now that you know what factors are, let us see the meaning of a prime number. A prime number is a number, which only has two factors: 1 and the number itself. 2 is a prime number, and 4 is not a prime number. This is because, 2 = 2 x 1 but, 4 = 2 x 2. So, 2 can only have two factors, itself and 1.
Also, the numbers which are a combination of prime numbers are called composite numbers. So, 4 is a composite number whereas 2 is a prime number (in fact, 2 is the only prime number which is even. All other even numbers are divisible by 2.)
What are the Prime Factors or Prime Factors Definition
Prime factors are simply factors of a number which are prime numbers. Here is an example:
Take the number 8. 8 can have different factors, that is, 8 = 4 x 2 = 2 x 2 x 2 = 8 x 1
So, we can say that the numbers 4, 2, 8, 1 are the factors of 8. But the prime factor of 8 is only 2 (because 1 is neither prime nor a composite number). 4 in itself is not a prime number (4 = 2 x 2) and neither is 8 (8 = 4 x 2).
So, prime factor meaning tells us that a number which is both prime and a factor of any given number is a prime factor.
How to Find Prime Factors of a Number
The method of finding prime factors of any given number is called prime factorization. What is the meaning of prime factorization?
Prime factorization is the process in which we write any number in the form of its prime factors. Now there are two different ways to find the prime factors of a number; the Factor tree method and the repeated division method.
How To Find Prime Factorization of A Number Using the Factor Tree Method
The method of factor tree is straightforward. Take the number, and then, with two arrows representing the branches of a tree, you break the number into any of the factors. You do this process until you reach the prime factors.
Example 1:
Find the prime factors of 36 using factor tree method.
Solution:
We see that the prime factors of 36 are 2 and 3. So, we can write the prime factorization of 36 as:
36 = 2 x 2 x 3 x 3.
One benefit of using the factor tree is that this method is graphical in how it represents the breakdown into factors. However, reaching the prime factors can be time-consuming.
How to Calculate Prime Factors using the Repeated Division Method?
As the name suggests, in this method, we begin by dividing the number with its smallest prime factor, and we keep on dividing the resultant quotient with its smallest prime factor until we reach 1 as the final quotient. Here’s an example.
Finding prime factors of 36 using repeated division method:
In this case, we divide 36 by the prime factor of 2 and get the resultant quotient 18, which we divide with 2 again. We keep on dividing until we get the number 1.
So, 36 = 2 x 2 x 3 x 3
The prime factors of 36 are 2 and 3.
FAQs on Prime Factors: Step-by-Step Explanation & Examples
1. What are prime factors? Can you give an example?
Prime factors are the prime numbers that you multiply together to get another number. For a number to be a prime factor, it must satisfy two conditions: it must be a factor of the original number, and it must be a prime number itself. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Out of these, the prime numbers are 2 and 3. So, the prime factors of 12 are 2 and 3. We can write 12 as a product of its prime factors: 12 = 2 × 2 × 3.
2. What is the difference between a factor and a prime factor?
A factor is any number that divides another number exactly, without leaving a remainder. For example, the factors of 20 are 1, 2, 4, 5, 10, and 20. A prime factor, however, is a factor that is also a prime number. From the list of factors for 20, only 2 and 5 are prime numbers. Therefore, while 4 and 10 are factors of 20, they are not its prime factors.
3. Which method is better for prime factorisation: the factor tree or repeated division?
Both the factor tree and repeated division methods will give you the correct set of prime factors. However, their suitability depends on the number.
- The factor tree method is very visual and great for smaller numbers, as it clearly shows how the number breaks down.
- The repeated division method (or ladder method) is often more efficient and organised for larger numbers. It keeps all the prime factors neatly listed on one side, reducing the chance of missing one in a complex tree.
4. How do you find the prime factors of a number using the repeated division method?
To find prime factors using repeated division, follow these steps:
- Start by dividing the given number by the smallest prime number that divides it exactly. For even numbers, this is always 2.
- Write the prime factor on the left and the quotient below the original number.
- Continue dividing the quotient by the smallest possible prime number.
- Repeat this process until the quotient becomes 1.
- The prime factors will be all the numbers you used to divide on the left.
5. How do you find the smallest prime factor for any given number?
To find the smallest prime factor of a number, you should test for divisibility by prime numbers in ascending order, starting with 2.
- First, check if the number is even. If it is, then 2 is its smallest prime factor.
- If it is not divisible by 2, check if it's divisible by 3 (sum of digits is divisible by 3).
- If not, check for divisibility by 5 (if it ends in 0 or 5).
- Continue this process with the next prime numbers (7, 11, 13, etc.) until you find one that divides the number exactly.
6. Why is 1 not considered a prime factor?
The number 1 is not considered a prime factor because it is not a prime number. A prime number is defined as a natural number greater than 1 that has exactly two distinct factors: 1 and itself. The number 1 only has one factor, which is 1. Including 1 as a prime factor would also violate the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 has a unique prime factorisation. If 1 were a prime factor, you could have infinite factorisations (e.g., 6 = 2×3, 6 = 1×2×3, 6 = 1×1×2×3, etc.).
7. Can a number have more than one set of prime factors?
No, a number cannot have more than one set of prime factors. This principle is known as the Fundamental Theorem of Arithmetic. It states that every integer greater than 1 is either a prime number itself or can be represented as a unique product of prime numbers. The order of the factors might change (e.g., 30 = 2 × 3 × 5 or 3 × 5 × 2), but the set of prime factors {2, 3, 5} will always be the same for the number 30.
8. How is prime factorisation used in other Maths topics like HCF and LCM?
Prime factorisation is a fundamental tool for finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of two or more numbers.
- To find the HCF, you multiply the lowest power of all the common prime factors between the numbers.
- To find the LCM, you multiply the highest power of all prime factors that appear in any of the numbers.
9. What is the easiest way to check if a large number is prime before trying to find its factors?
While there is no simple trick for very large numbers, the most practical approach for school-level problems is to use divisibility rules and then test division by prime numbers. First, quickly check the basic rules for 2, 3, 5, and 11. If none of these work, you need to test for divisibility by other prime numbers (7, 13, 17, 19, etc.). You only need to test primes up to the square root of the number. If you find no prime factor up to its square root, the number itself is prime.
10. Do prime factors always have to be small numbers?
No, prime factors are not always small. While many numbers are broken down into small primes like 2, 3, and 5, a number can have very large prime factors. For example, the number 217 is not divisible by 2, 3, or 5. Its prime factors are 7 and 31 (217 = 7 × 31). Similarly, a large prime number itself, like 997, has only one prime factor: 997. The size of the prime factors depends entirely on the composition of the original number.
