Unit 3 sec 1.4

Unit 3 sec 1.4 Prime factors

05 November 2015

21:24

Natural number greater than 1 that is not prime number is called composite number.

Unlike prime number, a composite number can be written as product obtained factors, neither of which is 1.

 composite numbers can often be written as products of even more factors.

So 360 can be written as a product of 4 factors, none of which is one

 360 = 4×5×3×6.

The process here can be set out as a factor tree.

    You can go hunting in the process until all the numbers at the end of the tree are prime numbers.

   Here, the number 360 is written as a product of prime factors. You can use a process similar to the one above to write any composite number as product of prime factors.

    The process of writing natural number as a product of factors greater than 1 (whether prime or not) is called factorisation. The factorisation above was started by writing 360 = 20×18, but it could also have been started by writing 360 = 10×36, or 360 = 9×40, for example. And there are different ways to proceed with the numbers further down the factor tree, too.

 In fact, no matter how you factorise a composite number into a product of prime factors, you will always obtain the same answer.

The fundamental theorem of arithmetic

every natural number greater than 1 can be written as a product of prime numbers in just one way (except that the order of the brines in the product can be changed).

  The prime factorisation of a natural number is the product of prime factors that is equal to it. Here are the prime factorisation is of the numbers from 2 to 10:

   2 = 2      3 = 3         4 = 2²        5 = 5         6 = 2 × 3            7 = 7

   8 = 2³     9 =3²         10 = 2 ×5.

The fundamental theorem of arithmetic is the reason why the prime numbers can be thought of as the building blocks of the natural numbers.

   When you want to write a number as a product of prime factors, it is sometimes helpful to be systematic. Start by writing the number as small as possible prime × a number, and do the same with each composite number in the factor tree. If you do this for the number 252, then you obtain the factor tree so 252 = 2² × 3² ×7.

  Round the systematic method, at each level of the factor tree Newgate prime factor and a composite factors. At each stage ‘the smallest possible prime’is the same as, or bigger than, the previous prime factor stop. You do not need to set out the working as a factor tree - you might say it is said to out like this:

252 = 2×126

       = 2×2×63

       = 2×2×3×21

       = 2 ×2× 3×3×7

       = 2²× 3² × 7.

   The method suggested above 12 finding the prime factorisation of a number works well the prime factors. There was small, but it is time - consuming if they are large. Despite much research, no one has managed to find a quick method for finding large prime factors of a number. So multiplying to large prime numbers together is a process that is collected to carry out but slow to reverse.

  A computer can multiply 150 - digit prime numbers in seconds, but a suitable computer program were given the result, then it would probably not be able to find the 2 prime factors within the human lifetime. Mathematicians have found a clever way to exploit this fact to design secure encryption systems. The security of personal information such as account details and credit card numbers has become an essential field of computer science.

 You can use prime factorisation to help you find the laws common multiple and highest common factors of sets of numbers.

Example 2 using prime factorisation to find LCMs and HCFs

find the laws common multiple and the highest common factor of 84 and 280 stop

solution

write out the prime factorisation, with a column for each different prime.

84 = 2² × 3    ×7

280 = 2³   × 5 × 7

to find the LCM, multiplying together the highest power of the prime column.

The LCM of 84 and 280 is

2³ ×3 × 5 × 7 = 280

to find the HFC, multiplying together the always power of prime in each column, considering only the prime that you are in all the rows.

The HFC of 84 and 280 is

2² × 7 = 28.

Strategy to find the LCM or HCF of 2 or more numbers

Ø find the prime factorisation of the numbers.

Ø To find the LCM, multiplying together the highest power of each prime factor occurring in any of the numbers.

Ø To find the HCF, multiplying together the lowest power of each prime factor common to all the numbers.

A method that can be useful when you want to find the lowest common multiple or highest common factor of just 2 numbers, which are fairly small.