How to Write a Number as a Product of Its Prime Factors
What Are Prime Numbers?
Before we can look at prime factors, we must first examine what a prime number is.
Prime numbers are defined as any number with exactly two factors (numbers which the original number will divide into perfectly with no remainder).
The number 2 is a prime number, as it has only two factors, 1 and 2. 5 is also a prime number, as it has exactly two factors, 1 and 5. In fact, for any prime number, its two factors must be 1 and itself.
On the other hand, the number 6 is not a prime number, as it can be divided by 1, 2, 3 and 6 for a total of four factors. 1 is also not prime, as it has only one factor: itself.
Prime numbers are defined as any number with exactly two factors (numbers which the original number will divide into perfectly with no remainder).
The number 2 is a prime number, as it has only two factors, 1 and 2. 5 is also a prime number, as it has exactly two factors, 1 and 5. In fact, for any prime number, its two factors must be 1 and itself.
On the other hand, the number 6 is not a prime number, as it can be divided by 1, 2, 3 and 6 for a total of four factors. 1 is also not prime, as it has only one factor: itself.
What are Prime Factors?
As the name suggests, the prime factors of a number are the factors of a number which are also prime. For example, the number 20 has six factors; 1, 2, 4, 5, 10 and 20. Of these factors, 2 and 5 are the only prime numbers; hence we call these the prime factors of 20.
When writing a number as a product of its prime factors, we want to rewrite the number as the multiplication of its prime factors which will get us back to this number. For example 20 = 2 × 2 × 5. We have three numbers, all of which are prime, which multiply together to make 20.
When writing a number as a product of its prime factors, we want to rewrite the number as the multiplication of its prime factors which will get us back to this number. For example 20 = 2 × 2 × 5. We have three numbers, all of which are prime, which multiply together to make 20.
A factor tree of 30
Using a Factor Tree for Prime Factorisation
When numbers have multiple factors, it can be tricky to find all of the prime factors. Furthermore, many numbers have repeated prime factors, such as in the previous example, where we saw that we used 2 twice to get 20. These can be especially difficult to spot.
One way to make sure that we find every prime factor needed is to use a prime factor tree. This works by taking our original number and breaking it down into a pair of factors that multiply together to give the original number.
If either of these factors is prime, we leave it there. If either number can be broken down again, we do so. We keep repeating this until we can go no further, at which point we will have every prime factor.
One way to make sure that we find every prime factor needed is to use a prime factor tree. This works by taking our original number and breaking it down into a pair of factors that multiply together to give the original number.
If either of these factors is prime, we leave it there. If either number can be broken down again, we do so. We keep repeating this until we can go no further, at which point we will have every prime factor.
Example: Write 140 as the Product of Its Prime Factors
The first step is to find two numbers that multiply together to make 140. There are multiple pairs that will do this, e.g., 2 × 70, 4 × 35, 5 × 28, etc., but it doesn't matter which one we choose. We will find out more about the reason why later on.
In this example, let's start with 10 × 14. To write this, we start with 140 at the top and have two branches coming down to our two factors, as in the picture below.
In this example, let's start with 10 × 14. To write this, we start with 140 at the top and have two branches coming down to our two factors, as in the picture below.
Prime factorisation tree for 140 - step one
Continuing the Prime Factorisation of 140
Now that we have our first pair of factors, we look at each factor in turn and split them up further if we can.
On the left, we have 10. This is equal to 2 × 5, so we split this along two branches to the 2 and 5. On the right, we have 14, which is equal to 2 × 7, and so we split this accordingly. The four factors which we have now are all prime, so we circle them to denote this and cannot split any further.
On the left, we have 10. This is equal to 2 × 5, so we split this along two branches to the 2 and 5. On the right, we have 14, which is equal to 2 × 7, and so we split this accordingly. The four factors which we have now are all prime, so we circle them to denote this and cannot split any further.
Prime factorisation tree for 140 - step two
Writing the Prime Factors as a Product
Our prime factor tree for 140 has four branch ends where we have a prime number and hence cannot split the factors any further. These are 2, 5, 2 and 7. Remembering that each pairing multiplies to make the number above, we can see that 140 must equal the product of these four prime numbers.
So we have 140 = 2 × 2 × 5 × 7 = 2^2 × 5 × 7
So we have 140 = 2 × 2 × 5 × 7 = 2^2 × 5 × 7
Can We Get Different Prime Factors?
Thinking back to our original split of the number 140, we had several options for pairs of factors. It might be tempting to think that if we had chosen a different starting pair, we would then end up with a different answer, but this isn't the case.
Look at the image below. In this case we start with 20 × 7. We circle 7 as this is prime, but continue to split 20. When we complete the tree we end up with the same four prime numbers as before; 2, 2, 5 and 7.
Look at the image below. In this case we start with 20 × 7. We circle 7 as this is prime, but continue to split 20. When we complete the tree we end up with the same four prime numbers as before; 2, 2, 5 and 7.
Prime factorisation tree for 140 - different starting pair
Unique Factorisation: The Fundamental Theorem of Arithmetic
In fact, it doesn't matter which order we choose to split up 140 or any other number into pairs of factors; we will always end up with the same prime factors at the end.
This is formally known as the Fundamental Theorem of Arithmetic which states that every integer larger than 1 can be written as a unique product of prime factors.
For example, 140 = 2^2 × 5 × 7 as we have already seen and cannot be written as a combination of any other primes.
1100 = 2^2 × 5^2 × 11 and again cannot be written as a product of any different primes or using different powers.
This is formally known as the Fundamental Theorem of Arithmetic which states that every integer larger than 1 can be written as a unique product of prime factors.
For example, 140 = 2^2 × 5 × 7 as we have already seen and cannot be written as a combination of any other primes.
1100 = 2^2 × 5^2 × 11 and again cannot be written as a product of any different primes or using different powers.
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