the prime numbers
factors
The factors of a number are all those numbers
which divide into it exactly*.
For example, the factors of 6 are 1, 2, 3 and 6
(because these are the numbers which exactly
divide into 6)
– whereas the factors of 7 are just 1 and 7
(because no other numbers divide exactly into 7).
*See our ‘What are factors?' for more material on
factors and how to introduce them.
Here are the factors of all numbers from 1 to 20 :
1
1
11
1, 11
2
1, 2
12
1, 2, 3, 4, 6, 12
3
1, 3
13
1, 13
4
1, 2, 4
14
1, 2, 7, 14
5
1, 5
15
1, 3, 5, 15
6
1, 2, 3, 6
16
1, 2, 4, 8, 16
7
1, 7
17
1, 17
8
1, 2, 4, 8
18
1, 2, 3, 6, 9, 18
9
1, 3, 9
19
1, 19
10
1, 2, 5, 10
20
1, 2, 4, 5, 10, 20
prime numbers
As you can see, some numbers have two factors,
some have three factors, some have four factors
and so on. A number which has exactly two factors
is called a prime number. In our list above the prime
numbers are 2, 3, 5, 7, 11, 13, 17 and 19.
Whole numbers which aren’t prime numbers are
called composite numbers or sometimes rectangle
numbers (for obvious reasons, eg 12 can be shown
as a 3 x 4 rectangle, 18 can be shown as a 3 x 5
rectangle and so on). 1 hasn’t got two factors so
clearly it isn’t a prime number.
Why are prime numbers important? For one thing,
prime numbers are like building blocks for all the
other numbers. Take any number you like and you’ll
find it’s either a prime number – or else it can be
written as a product of prime numbers, like these :
12
=
2 x 2 x3
15
=
3 x 5
38
=
2 x 19
42
=
2 x 3 x 7
56
=
2 x 2 x 2 x 7
100
=
2 x 2 x 5 x 5
Here’s a way of generating the first few prime
numbers; it’s called the
Prisoners Rocky Island Maths Experiment
Early in the 1900s the prison system in America
had a problem : many of the toughest prisoners
kept escaping. To solve this problem they built a
new gaol on Rocky Island, off the western shore of
the USA. Here they housed the most difficult and
desperate criminals. The prisoners were kept in
separate huts, numbered 2, 3, 4, 5 . . . and so on.
(There was no number 1 as the prison governor
didn’t want any prisoner to be able to say, ‘I’m the
number one!’)
Here’s a picture of the huts on Rocky Island :
The palm trees were planted to give the prisoners
something cheerful to look out on – the rest of the
island was all sand and rock.
When it was exercise time for the prisoners the
governor knew that it would be too dangerous just
to open all the huts at once, so he decided on an
experiment. The experiment was called the
‘Prisoners Rocky Isand Maths Experiment’ and this
is how it works :
The first hut is number 2, so keep hut 2 closed – but
open all the huts numbered with multiples of 2,
that’s to say 4, 6, 8, 10, 12 . . . and so on. This is
how things will look once you’ve done this :
The next hut with its number showing is hut number
3, so keep hut 3 closed – but open all the huts
numbered with multiples of 3, that’s to say 6, 9, 12,
15 . . . and so on. (Obviously some have been done
already.) This is how things will look now :
The next hut with its number showing is number 5,
so keep hut 5 closed – but open all the huts
numbered with multiples of 5, that’s to say 10, 15,
20 . . . and so on. Now this is the picture :
What’s special about the numbers which are left?
They are the prime numbers.
So Prisoner’s Rocky Island Maths Experiment was
a good title. To help everyone remember the first
few prime numbers (those below 50), the prisoners
on Rocky Island wrote a song; this is how it goes :
Two, three, five, seven – Rocky Island’s just like
heaven
Eleven, thirteen, seventeen – the nicest place
you’ve ever seen
Nineteen, twenty-three, twenty-nine – the guards
are all great friends of mine
Thirty-one, thirty-seven, forty-one – this is the place
the sun shines on
Forty-three, forty-seven – Oh, what a crime!
Locking up numbers, just ‘cos they’re PRIME
notes
o If you decide to use the Rocky Island
presentation with your pupils, you’ll find it’s
visually more effective to draw the huts in one
long line – and it’s even better if you can go a
little further than we do (up to 35, say).
o There is no known pattern or formula for
generating the prime numbers.
Finally, here are all the prime numbers up to 1000 :
PRIME NUMBERS TO 1000
2 3 5 7 11 13 17 19 23
43 47 53 59 61 67 71 73
101 103 107 109 113 127
149 151 157 163 167 173
193 197 199 211 223 227
241 251 257 263 269 271
293 307 311 313 317 331
353 359 367 373 379 383
409 419 421 431 433 439
461 463 467 479 487 491
521 523 541 547 557 563
587 593 599 601 607 613
641 643 647 653 659 661
691 701 709 719 727 733
757 761 769 773 787 797
823 827 829 839 853 857
881 883 887 907 911 919
947 953 967 971 977 983
29 31 37 41
79 83 89 97
131 137 139
179 181 191
229 233 239
277 281 283
337 347 349
389 397 401
443 449 457
499 503 509
569 571 577
617 619 631
673 677 683
739 743 751
809 811 821
859 863 877
929 937 941
991 997