
sums will set you freehow to teach your child numbers arithmetic mathematicsprime numbers and factors, 

prime numbers and factors, the sieve of EratosthenesA prime number is a number (other than one) that is divisible only by 1 and by itself [1]. Its factors are then one and that prime number. A prime number is also a number that cannot be divided any further. However, the number one needs to be paid little attention as multiplying any integer[2] by one leaves the integer unchanged. Thus two times one equals two (2 x 1 = 2), one hundred and fiftyseven multiplied by one equals one hundred and fiftyseven (157 x 1 = 157), and one million, eight hundred and twentyfour thousand four hundred and one multiplied by one remains that big number (1,824,401 x 1 = 1,824,401). Dividing by one, likewise, leaves the integer unchanged. So one hundred and fortythree divided by one remains as one hundred and fortythree (143 ÷ 1 = 143). Fortytwo times one times one times one still remained fortytwo (42 x 1 x 1 x 1 = 42), as does fortytwo times one times one times one divided by one (42 x 1 x 1 x 1 ÷ 1 = 42). Caution: this behaviour tends to come unstiched with the ‘number’ zero, otherwise known as nothing. Doubtless, zero will raise its ugly little nonexistent head again in due course. The factors (of a number) are the prime numbers that multiply up to make the number concerned. For instance, the number 10 has the factors 1, 2 and 5. That is, one times two times five makes ten. 1 x 2 x 5 = 10. But from now on, we shall ignore the one, and so we list the factors of 10 as 2 and 5, or 2 by 5, 2 x 5. Another example is the factors of 12, which are 2, 2, 3, as in 2 x 2 x 3 = 12.
the sieve of Eratosthenes [2]Here follows the method of Eratosthenes’ sieve to determine prime numbers up to 200. Of course, if you wish, you can continue and determine prime numbers of greater value than those shown here.
With an Eratosthenes’ sieve, the multiples of each prime number are progressively crossed out of the list of all numbers being examined (in this case the numbers one to two hundred, 1 to 200). You will notice that by the time you come to crossing out the multiples of three, several have already been crossed out: 6, 12, 18 etc. The blank sheet can, of course, be used for an exercise in crossing out other series alone, such as just threes, as well as nonprimes, such as fours or sixes. an example of using an Eratosthenes’ sieveThe first prime number other than one is two (2). All multiples of two are crossed off the table (here we have used red for two and its multiples). Thus the sieve has removed all the nonprimes which are divisible by the prime number two. The sieve removes nonprimes and what will eventually be left in the sieve will be the remaining prime numbers. 


The next prime number is three (3). Here its multiples are crossed off in blue, unless they have already been crossed out as a multiple of two. Now the sieve has removed all the nonprimes which are divisible by the prime numbers two and three. Thence the process continues. Five is the next prime number. Multiples of five are crossed off in green, unless they have already been crossed out. The next prime number is seven. Its multiples are crossed out in orange, unless previously crossed. Below, all further prime numbers have been outlined in purple. The purple crossed out numbers are multiples of prime numbers greater than seven. You will see that, when you come to the seventeens (17), all nonprimes have already been crossed out [34, 51, 68, 85, 102, 119, 136, 153, 170, 187]. That leaves the primes uncrossed and they are highlighted above with coloured boxes. Of the ’teen numbers before seventeen, note that 13 x13, 13^{2}, is already 169, while 13 x 14 is 182, 13 x 15 is 195, and 13 x 16 exceeds 200, which is as far as this table goes. 13 x 17 is 221, well beyond our table’s upper limit. 14, 15, 16 have already been crossed off the table because they are divisible by earlier prime numbers. (14 = 2 x 7, 15 = 3 x 5, 16 = 2 x 2 x 2 x 2.) The full list of prime numbers from 1 to 200 is:
LCM  lowest common multipleTo understand the idea of Lowest Common Multiple  LCM, first you need you understand what is a multiple. When a particular whole number is multiplied by any other whole number, the resulting number (the result) is a multiple of the number you first thought of. For example, with the number three (3),
3 is its first multiple because 3 x 1 = 3. This list of multiples, and the following multiples of
three, can be seen easily using a multiplication
cross table.
common multiplesThe shared or common multiples of two numbers are the numbers which are multiples of both numbers. To find the common multiples of, say, 8
and 12,
you first list multiples of each number: On the following cross table, common factors of 8 and 12 are highlighted in crimson.
So common multiples of 8 and 12 are 24, 48, 72, 96 and so on. The lowest common multiple is the smallest multiple of a pair, or a group, of numbers. So for the pair of numbers, 8 and 12, the lowest common multiple is 24. The LCM can also be seen as the smallest number into which both (or all) numbers will divide.
HCF  highest common factorTo understand the idea of Highest Common Factor  HCF, first you need you understand what is a factor. As was explained previously, the factors (of a number) are the prime numbers that multiply up to make the number concerned. For instance, the number 10 has the prime factors 1, 2 and 5. That is, one times two times five make ten. 1 x 2 x 5 = 10. We shall continue to ignore the one, and so we list the prime factors of 10 as 2 and 5, or 2 by 5, 2 x 5. common factorsWhen two or more numbers have the same factor, that factor is called a common factor. For instance, to find the common factors of 8 and 12, you first list the prime factors to be found in each of those numbers:
Now, the highest common factor
is the biggest number that will divide into
a pair, or a group, of numbers, without leaving a remainder. As you can see in the example above, the common prime factors of 8 and 12 are 2 and 2. Note carefully, both our original numbers, 8 and 12, have in common two lots of the factor 2. When multiplied together, two times two equals four, 2 x 2 = 4, these prime factors make a composite common factor: 4. So the highest common factor of 8 and 12 is 4. However, to find the highest common factor of two or more numbers, some people prefer to use the method of looking for all the factors of those numbers, not just the prime factors. But you must be aware, that although this method works well with smaller numbers that have fewer and smaller factors, it is hopeless for larger numbers which you cannot easily deconstruct into all their factors. For those larger numbers, it will be necessary to return to basics, as described above, and determine the prime factors concerned, and then multiply those back up as necessary. Using the method of looking for all the factors in a number:
Thus, the common factors of 8 and 12 are 2 and 4, so the highest common factor of 8 and 12 is 4.
end notes


sums will set you free includes the series of documents about economics and money at abelard.org.  
moneybookers information  egold information  fiat money and inflation  
calculating moving averages  the arithmetic of fractional banking  
You are here:
how to teach your child
number arithmetic mathematics  prime numbers and factors, the sieve of Eratosthenes < sums will set you free < Home 
email abelard at abelard.org © abelard, 2009,12 september the web address for this page is http://www.abelard.org/sums/teaching_number_arithmetic_mathematics_factors_primes.php 