• Understanding prime numbers and why they are so important
• Being able to tell when one number is divisible by another
• Knowing how to break a number down into its basic components
Odd numbers, even numbers, prime numbers, composite numbers, square numbers, … these are just a few of the different types of numbers that mathematicians get incredibly excited about. What are all these different sorts of number? Most of these terms refer to the different ways that whole numbers are built out of others. This will become clearer when we have met the most important numbers of all: prime numbers.
The definition of a prime number is simple: a prime number is a whole number which cannot be divided by any other whole number (except 1 and itself). So, for example, 3 is prime because the only way to write 3 as two positive whole numbers multiplied together is as 3 × 1 (or 1 × 3, which is essentially the same thing). On the other hand 4 is not prime because 4 = 2 × 2.
A composite number essentially means a “non-prime” number, and 4 is the first example. Similarly 5 is prime, but 6 is composite. (The numbers 0 and 1 are so special that they deserve categories of their own, and are classed as neither prime nor composite.)
The first 25 primes are: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
It was Euclid, in around 300 BC, who first proved that the list of primes goes on forever. There is no largest prime number, and so people keep finding bigger and bigger ones. It is a tough job though, as telling whether a very large number is prime or composite is hard. The largest prime known so far is 12,978,189 digits long!
Why do people get so excited about prime numbers? The reason they are so important is that they are the fundamental blocks from which all other numbers are built. Although 6 is not prime, it can be broken down into primes as 3 × 2. Similarly 8 can be broken down as 2 × 2 × 2, and 12 as 2 × 2 × 3. In this sense, prime numbers are like mathematical atoms: everything else is built from them.
BREAK SOME NUMBERS DOWN INTO PRIMES IN QUIZ 1.
What is more, this chapter’s golden rule says a little bit more than this. Not only can every number be broken down into primes, but there is only one way to do it. So once we know that 1365 = 3 × 5 × 7 × 13, for example, it follows that the only other ways to write 1365 as a product of prime numbers are reorderings of this: 5 × 3 × 13 × 7, for example. So we know automatically, without having to check, that 1365 ≠ 5 × 5 × 5 × 11 (the symbol ≠ means “is not equal to”). This rule goes by the grand title of The fundamental theorem of arithmetic.
Even numbers are those which appear in the two times table: 2, 4, 6, 8, 10, … Another way to say the same thing is that even numbers are those which have 2 as a factor, meaning that 2 can divide into the number exactly, without leaving a remainder. Yet another way to say the same thing, is that the even numbers are the multiples of 2.
Odd numbers, of course, are the remaining numbers: the numbers which do not have 2 as a factor.
Factor and multiple are opposite terms. To say that 15 is a multiple of 3 is the same as saying that 3 is a factor of 15. Both statements mean that 3 can divide into 15 exactly, without leaving a remainder. In other words, 15 is in the three times table.
TRY OUT THESE TERMS IN QUIZ 2.
It is often useful to know whether or not a large number is a multiple of a particular smaller number. For some small numbers this is so easy that we can do it without thinking:
• The multiples of 2 are exactly the even numbers, meaning all the numbers that end in 2, 4, 6, 8 or 0.
• The multiples of 5 are the numbers that end in a 5 or a 0, such as 75 and 90.
• The multiples of 10 end in 0s, such as 80, 250, 16,700.
For other small numbers there are other tests, which are slightly subtler:
• You can tell whether or not a number is a multiple of 3 by adding up its digits. If the total is a multiple of 3, then so was the original number. So 117 is a multiple of 3, because 1 + 1 + 7 = 9, which is a multiple of 3. On the other hand 298 is not a multiple of 3, because 2 + 9 + 8 = 19.
• A number is a multiple of 6 if it passes the tests for 2 and 3. So 528 is divisible by 6, since it is even, and 5 + 2 + 8 = 15, which is divisible by 3. (Notice that the total of the digits does not have to be divisible by 6.)
• The test for divisibility by 9 is similar to the test for 3: add up the digits, and if the result is a multiple of 9, then so was the original number. So 819 is a multiple of 9, since 8 + 1 + 9 = 18, but 777 is not, since 7 + 7 + 7 = 21.
• You can tell whether a number is a multiple of 4 just by looking at its last two digits. If they are a multiple of 4, then so is the whole thing. So 116 is a multiple of 4, just because 16 is. Similarly 5422 is not a multiple of 4, as 22 isn’t.
• The number 8 is a little awkward, and there are various possible ways forward. One is a variation on the test for divisibility by 4. (Another is to give up and use a calculator!) If the last three digits of the number are divisible by 8, then so is the original number. So 6160 is divisible by 8, since 160 is. The trouble is that telling whether a three-digit number is divisible by 8 is not something most people can do on sight. The best option is to divide the three-digit number by 2, and then apply the test for divisibility by 4. So if we want to know whether 7476 is divisible by 8, first take the last three digits (476) and then divide by 2 (238) and finally look at the last two digits of that (38). In this case that is not a multiple of 4, so the number fails the test.
PRACTISE THESE TESTS IN QUIZ 3!
• The fiddliest single-digit number is 7. There is a workable test though, and it goes like this. To test 399 for divisibility by 7, chop off the last digit (9) and double it (18). Then subtract that from the truncated number (39 − 18 = 21). If the result is divisible by 7, then so is the original number, which in this case it is. With this test we might end up with 0: for instance if we apply the test to 147, we get 14 − 14 = 0. In this situation, 0 does count as a multiple of 7, and so the number passes the test.
• The number 11 has a lovely test! It goes like this. Go through the digits, alternating between adding and subtracting. If the result is divisible by 11, then so is the original number. To test 9158, we go 9 − 1 + 5 − 8 = 5, which is not divisible by 11, so the test is failed. It’s possible to end up with 0 again, or even negative numbers, but that’s no problem. We do count 0 and −11, and −22, and so on, as multiples of 11. So 1914 is a multiple of 11 since 1 − 9 + 1 − 4 = −11 is divisible by 11.
BREAK SOME LARGER NUMBERS DOWN IN QUIZ 4.
Earlier in the chapter, we said that every number can be broken down into primes, and we saw some examples. But if we are given a larger number, such as 308, how can we actually find out what its prime ingredients are? The idea is to try dividing by prime numbers in turn, using the tests we’ve just seen. To start with, 308 is undoubtedly even. So we can divide it by 2, this leaves 154. This is also even, so we can divide it by 2 again, to get 77. Now, this is no longer even, so we exhausted the 2s, and we move on to the next prime. We might try dividing 77 by 3, but it fails that test. It is also easy to see that 77 is not divisible by 5. So the next prime on the list is 7, and 77 is indeed divisible by 7. Dividing it by 7 leaves 11, which is itself prime. So we have finished. Collecting together all the primes that we divided by, we get: 308 = 2 × 2 × 7 × 11.
The prime numbers are as mysterious as they are important, even today. If you look at the sequence of prime numbers, there seems to be very little order to it. Sometimes primes come very close together, like 11 and 13, and sometimes there are larger gaps such as between 199 and 211.
TRY GOLDBACH’S CONJECTURE FOR YOURSELF IN QUIZ 5.
There are lots of seemingly basic facts about the prime numbers that we still do not know for sure. One of these is Goldbach’s conjecture. In 1742, Christian Goldbach noticed that every even number from 4 onward is actually the sum of two prime numbers. So 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, … If you can prove that Goldbach’s conjecture must be true for every even number, then you will have outshone the mathematicians of the last two centuries. Although it has been verified up to an enormous limit (around 1018—see The power of 10 for what this means), no-one has yet managed to prove that it must be true for all even numbers.
Sum up To get to know a number well, you need to know which other numbers divide into it. The most important ones to check are the atoms of the mathematical world, the primes!
1 Break these numbers down into primes.
a 15
b 18
c 21
d 24
e 32
2 Which are true and which are false?
a “18 is a multiple of 3”
“18 is a factor of 3”
b “246 is a multiple of 5”
“5 is a factor of 246”
c “4 is a multiple of 108”
“108 is a factor of 4”
d “114 is a multiple of 6”
“6 is a factor of 114”
e “245 is a multiple of 7”
“7 is a factor of 245”
3 Test these numbers for divisibility up to 11.
a 64
b 42
c 75
d 176
e 68
4 Break these numbers down into primes.
a 30
b 210
c 108
d 189
e 1617
5 Goldbach’s conjecture! Write these even numbers as two primes added together.
a 10
b 12
c 14
d 16
e 18