Product Of Prime Factors Video
A factor tree can be used to express a number as a product of prime factors. Make sure you know your first 5 prime numbers before you start these examples. These are 2,3,5,7 and 11 (a prime number can only be divided by 1 and itself) as these will come in handy. On some harder question you may need to know more prime numbers.
Write down 140 as a product of prime factors.
First think of two numbers that multiply together to give 140.
Let’s go with 2 × 70.
2 is a prime number (so circle it) and 70 is not a prime number, so 70 needs to be split into another pair of factors.
Let’s write 70 as 7 × 10.
Now, 7 is a prime number and 10 is not prime. Circle the 7, and split the 10 into another factor pair.
10 can be written as 2 × 5. 2 and 5 are both prime numbers so circle the 2 and 5.
Now all the numbers that we have circled are prime factors of 140.
So the product of prime factors is:
2 × 2 × 5 × 7.
This can also be written using powers as 2² × 5 × 7.
You can also check that the final answer is correct by mutlpying the numbers together. 2 multiplied by 2 is 4, 4 multiplied by 5 is 20, and 20 multiplied by 7 gives 140. This is why its called the product of prime factors as product means multiply, and all the prime factors multiply to give the number in the question.
Another good tip is to circle the prime numbers in a different coloured pen so that the prime numbers stand out.
Write down 252 as a product of prime factors.
First think of two numbers that multiply together to give 252.
Let’s go with 2 × 126.
2 is a prime number (so circle it) and 126 is not a prime number, so 126 needs to be split into another pair of factors.
Let’s write 126 as 2 × 63.
Now, 2 is a prime number and 63 is not prime. Circle the 2, and split the 63 into another factor pair.
63 can be written as 7 × 9. 7 is a prime number (so circle it) and 9 is not a prime number, so 9 needs to be split into another pair of factors.
9 can be written as 3 × 3. 3 is prime so circle both 3’s.
Now all the numbers that we have circled are prime factors of 252, and so the product of prime factors is:
2 × 2 × 3 × 3 × 7 or 2² × 3² × 7 using indices (powers).
Let's check the answer is correct now. 2 multiplied by 2 gives 4, 4 multipied by 3 gives 12, 12 multiplied by 3 gives 36, and 36 multipied by 7 gives 252 which is the number we are looking for.
Although factor trees are a good way to work out the product of prime factors, you can work out the product of prime factors without a factor tree. You can do this by dividing the number by the smallest prime number possible, and keep repeating the process.
So let's take a look at one last example without using a factors tree.
Write down 450 as a product of prime factors.
First divide 450 by 2 to give 225 (2 is the smallest prime factor).
Now divide 225 by 3 to give 75 (3 is the next small prime factor as 2 doesnt go into 225).
Now divide 75 by 3 again to give 25 (3 again is the next smallest prime factor).
Finally divide 25 by 5 to give 5 (5 is the only prime to go into 25).
So you can write all these prime factors down (including the last one) to give a final answer of 2 x 3 x 3 x 5 x 5. If you want this in index form then you can write this as 2 multiplied by 3 squared multiplied by 5 squared. Only do this though if the question asks as you could make an error.
Again, check the final answer to see if it's correct. 2 multipied by 3 gives 6, 6 multiplied by 3 gives 18, 18 multiplied by 5 gives 90, and 90 multiplied by 5 is 450.
This second method can be quicker than using factor trees, but it can be harder if you struggle with your times tables. Both methods shown here will give you the correct answer though.
Another thing to mention is that there can be different factor trees for each number, but all will result in the same answer. Once you have mastered the technique, then you can use the product of prime factors to work out the highest common factor and lowest common multiple of a pair of numbers. This can be done by working out the product of prime factors for each number and putting the number in a Venn diagram.