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Prime Factors

A number which has no factors other than ‘1’ and ‘the number itself’ is called a prime number

So; 2, 3, 5, 7, 11, 13, 17, 19, 23, … are prime numbers

Numbers that are not prime are called composite

So ‘4‘ is composite (it has ‘2’ as a factor) and so is ‘6‘ (it has ‘2’ and ‘3’ as factors)

So are; 8, 9, 10, 12, 14, 15, 16, 18, 20, …

i.e:

	      prime   prime	      prime	      prime			      prime
		?	?		?		?				?
		?	?		?		?				?
	1	2	3	4	5	6	7	8	9	10	11	12
	?			?		?		?	?	?		?
     neither			?		?		?	?	?		?
			    composite	    composite	  composite composite composite	    composite

A composite number can be “broken down” into its prime factors:

 

e.g. Find the prime factors of ‘168’

Working through the prime numbers in order (so starting with ‘2’), see which one divides into the number.

In this case, the first prime number, ‘2‘, does divide into our number:

		   ??? number
		   ?
 divisor ??????	2|168
		   84 ??? quotient

But ‘2‘ will divide into this again:

		2|168
		 2|84
		   42

And, yet again, ‘2‘ is still a factor:

		2|168
		 2|84
		 2|42
		   21

Now, ‘2’ is no longer a factor, so we move onto the next prime number, ‘3‘; which is a factor:

		2|168
		 2|84
		 2|42
		 3|21
		    7

Now that we have reached a prime number, there is no point in dividing anymore. Since, our number (‘168’) was divided by the prime numbers; 2, 2, 2 and 3, leaving a prime number of 7 as the final quotient, these are the prime factors of 168:

		  168 =  2 ×  2 × 2 ×  3 × 7

Or, to simplify it, we can write:

		  168 = 23 × 31 × 71

Question 2, part (a): Starting with ‘2‘, which is a factor of ’18’:

		2|18
		   9

Now ‘3‘ will divide into this:

		2|18
		 3|9
		   3

Now that we have a quotient of ‘3‘ (which is a prime number), there is no point in dividing anymore. Since, our number (’18’) was divided by 2 and 3, leaving a prime of ‘3‘ as the final quotient, these are the prime factors of 18:

		  18 = 2 × 3 × 3

Or, to simplify it, we can write:

		  18 = 21 × 32

Question 2, part (b): Starting with ‘2‘, which is a factor of ’30’:

		2|30
		  15

Now ‘3‘ will divide into this:

		2|30
		3|15
		   5

Now that we have reached a quotient of ‘5‘ (which is prime), there is no point in dividing anymore. Since, our number (’30’) was divided by 2 and 3, leaving a prime of 5, these are the prime factors of 18:

		30 = 2 × 3 × 5

Question 3: In question 2(a), we found the prime factors of 18 were:

		18 = 2 × 3 × 3

And in question 2(b), we found the prime factors of 30 were:

		30 = 2 × 3 × 5

Since:

		540 =   18    ×   30
		      ???????   ???????
		    = (2×3×3) × (2×3×5)

		    =    2×2×3×3×3×5

Which simplifies to: 540 = 22 × 33 × 51

 

Question 4: If we find the prime factors of ’40’:

Starting with ‘2‘:

		2|40
		  20

But ‘2‘ will divide again into this:

		2|40
		2|20
		  10

And again:

		2|40
		2|20
		2|10
		   5

Now that we have reached ‘5‘ (which is a prime number), there is no point in dividing anymore. Since, our number (’40’) was divided by 2, 2 and 2, leaving 5, these are the prime factors of 40:

		40 = 2 × 2 × 2 × 5

Which is the same as saying:

		40 = 23 × 5

So, n must be…

 

Question 5: First find the prime factors of 126, then you’ve easily be able to say what x is…

 

Question 6: First find the prime factors of 720…

 

Question 8: We already worked out the prime factors of 2520 and 126:

So, we know:			2520  =  23 × 32 × 51 × 71 
And, we also know: 		 126  =  21 × 32      × 71 

Which means:			2520  =  22 ×      51 
				 126

Which is easy to work out, even without using a calculator…

 

Question 9: This is a little tricky…

If your stuck, then let’s use a real example instead to work out what is going on:

Let’s imagine x = 288 (its prime factors are: 25 × 32) and y = 24 (its prime factors are: 23 × 31)

Then xy = 6912
What are the prime factors of 6912?
How do the powers relate to the prime factors of ‘288’ and of ’24’? (In particular, look at the POWERS)

Also, x/y = 12
How do the prime factors of 12 relate to the prime factors of ‘288’ and of ’24’?

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