It is usual to write the prime factorisation in index form with the primes in increasing order, Then use division to test divisibility by the remaining primes 7, 13, 17, 19, … up to.These divisions tests are presented in the module, Multiples, Factors and Powers. Use the standard divisibility tests to test n for divisibility by the primes 2, 3, 5 and 11.When testing whether a whole number n is prime, it is sufficient to test divisibility byĪny number n that is to be tested in school mathematics should thus yield to the following plan of attack:.When they sieved by the next prime 11, they would have found that all the multiples of 11 were already crossed out, because the first multiple of 11 not crossed out already would be 11 × 11 = 121, which is greater than 100. Students will have noticed from the Sieve of Eratosthenes that when finding all primes up to 100, it was only necessary to sieve by the primes 2, 3, 5 and 7. One important insight, however, greatly reduces the amount of computation required. Testing whether a reasonably large number is prime is a massive computing problem. This is intuitively clear once one observes that more and more prime numbers are used in the sieving process as the numbers get bigger. The table does give some hint that the primes become more widely spaced on average as the numbers get bigger. Apart from the obvious fact that all primes end in 1, 3, 7 or 9, except for 2 and 5, there are no other obvious patterns.
The patterns within the sequence of prime numbers are notoriously complicated, and have generated some of the most famous solved and unsolved problems in mathematics. The 25 circled numbers are the primes up to 100, and the 74 crossed-out numbers (not including 0 and 1) are the composite numbers up to 100. Continue until all numbers are either circled or crossed out.Circle the next number not crossed out (which is 3), and then cross out multiples of 3.Circle the next number 2, and then cross out every multiple of 2.Cross out 0 and 1 - they are exceptions.Write down all the whole numbers up to 100.Here is a systematic way of writing down all the prime numbers, and all the composite numbers, up to 100. It is better to be precise than to invite arguments about what may be implied, but unsaid. Students often realise that a prime number like 5 has other factors if larger number systems are considered, for example, The phrase ‘whole number factor’ makes the restriction in the definitions quite clear. The number system we are talking about is the set of whole numbers 0, 1, 2, 3, …, and The phrase also excludes 0, which is divisible by every whole number and so has no sensible prime factorisation. The phrase ‘greater than 1’ is needed in the definition of composite numbers to exclude 1, which has no prime factors and so is not the product of two or more prime numbers. We do not want 1 to be a prime number, otherwise the factorisation of numbers into primes would not be unique. The phrase ‘greater than 1’ is needed in the definition of prime numbers to exclude 1. A composite number is a whole number greater than 1 that is not a prime number.A prime number is a whole number greater than 1 whose only whole number factors are itself and 1.’.Prime numbers and composite numbers need to be defined rather carefully: the composite numbers 4, 6, 8, 9, 10, …, which can be factored into the product.the prime numbers 2, 3, 5, 7, 11, …, which cannot be factored into smaller numbers,.We leave aside the numbers 0 and 1, and then organise the remaining whole numbers 2, 3, 4, 5, … into: The discussion above shows that for the purposes of prime factorisation, we need to distinguish three types of whole numbers. Prime factorisation is a very useful tool when working with whole numbers, and will be used in mental arithmetic, in fractions, for finding square roots, and in calculating the HCF and LCM. Every compound can be broken down uniquely into its elements, but if we are given the elements, there are often a great many different compounds that can be formed from them. In other situations, however, such processes do not work nearly as straightforwardly, as can be illustrated using the analogy of chemistry. (2 × 2) × (3 × 5) = 4 × 15 = 60 or (2 × 5) × (2 × 3) = 10 × 6 = 60Īnd we will always get the same original number, whatever order we choose for multiplying the prime factors. Conversely, if we are given the prime factors of a number, we can reconstruct the original whole number by multiplying the prime factors together, Thus we can factor any whole number into a product of prime numbers, for exampleĪnd this prime factorisation is unique, apart from the order of the factors. A fundamental technique in mathematics is to break something down into its component parts, and rebuild it from those parts.
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