Q. Explain why any number ending in 0 cannot be prime?


A. If the number ends in 0 it will be divisible by 2, 5, 10 and probably many other factors depending on the size of the number, certainly any number ending in a 0 has to be divisible by 10 and is automatically therefore excluded from being prime.


Q. Explain why any number ending in 4 cannot be prime?


A. Again, if a number ends in 4 it must be divisible by 2, so it is automatically not a prime number.


Prime numbers are fairly straightforward, what we will do now is take a look at another mathematical concept involving prime, that is prime factorisation. Prime factorisation is the means by which we write down numbers which are not prime, as a product of any prime factors the number possesses, usually this is done in what is called an "index tree" which is a bit like a family tree descending chart. At the top of the tree is written the number that you are going to "prime factorise" and then descending down the tree left and right will be the prime factors that that number possesses. Let's take a look at a couple of examples to make sure that we understand what this is actually saying to us.


Q. Write down the number given as a product of its prime factors - 68


A. The way we do this is to take factors starting at 2 (we don't start at 1 because 1 divides into any number) and we see if it will divide into the number, if it does, we write 2 just below the number to the left, and the result of the division just below the number to the right. We carry on using 2 until it won't divide any more, then we move to our next prime number which is 3, we repeat the process until we can no longer use 3 then we move to the next prime number 5, then 7 then 11 and so on. We only consider prime numbers hence the name prime factorisation.


Let's go back to number 68, 68  divides by 2 leaving 34, 34 divides by 2 leaving 17. 17 is itself a prime number therefore our prime factorisation of number 68 has finished. If we want to get back to 68 we take our prime factors and multiply them together - 2 x 2 x 17 = 68. This isn't the way we would leave the answer though, 2×2 would be represented in its index form as "2 squared" so we would in fact write our result as 22 x 17.


Q. Write down the number 325 as a product of its prime factors.


A. Does 325 divide by 2? Well it is an odd number therefore cannot divide by 2, so we move to our next prime number which is 3. does 325 divide by 3? Well there are some rules about whether a number will divide by a certain other number but we will come to these later, some of them are easy some of them are quite complicated so we won't worry ourselves at this point. Suffice it to say, 325 will not divide by 3 so we move on to our next prime number which is 5. 325 will divide by 5 leaving us with the result of 65. 65 again will divide by 5 which leaves us 13. We stop here because 13 itself is a prime number so the divisions can no longer continue. The result of our prime factorisation of 325 is 5 x 5 x 13 which will of course be written as 52 x 13.


Q. Write down the number 677 as a product of its prime factors.


A. 677 is an odd number therefore straight away we can exclude any attempts to divide it by 2, will 677 divide by 3? If you try this in the calculator you will end up with a recurring decimal as part of your result, therefore 677 does not divide by 3, neither does it divide by 5, 7, 11, 13, 17, 19 and so on in fact 677 is itself a prime number.


Q. Write down the number 679 as a product of its prime factors.


A. 679 is an odd number therefore cannot be divisible by 2, so we move straight away to 3, however, we find that it is not divisible by 3, nor 5. 679 is divisible by 7 leaving a result of 97, but 97 itself is a prime number therefore at factorisation can go no further and we write our answer as 7 × 97.


Q. Write down the number 55,296 as a product of its prime factors.


A. 55,296 is an even number therefore our first prime factor can be number 2. we divide 55,296 x 2 which leaves us with 27,648. Again this is an even number so we can once again divide by 2 which gives us 13,824, again we have an even number, we divide once again by 2 which gives us 6912 which again divides by 2 to give us 3456, which again divides by 2 to give us 1728 and this divides by 2 to give it 864 which divides by 2 to give us 432 which divides by 2 to give 216. 216 divides by 2 to give us 108, and 108 divides by 2 to give us 54. 54 divides by 2 to give us 27 and thankfully this is where the division by 2 comes to an end. 27 is an odd number which cannot divide by 2 but we can see that it divides by 3 leaving 9 which again divides by 3 to leave 3. 3 is a prime number therefore the division can go no further. Now we need to work out how many times we have divided by 2 and 3, you should be able to see that 55,296 is in fact 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 which we would represent as 211 x 33.


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