Number Theory: A Very Short Introduction

has the form 4n, and if b is odd then b² has the form .

An immediate consequence of this result is that none of the numbers

can be a square. This is because these numbers all have the form , for some integer n: for example, and .

But we can say a little more. We’ve just seen that if , then

But is the product of two consecutive integers (one odd and the other even), and so is even. It follows that is divisible by 8, and so:

The square of every odd number has the form , for some integer n.

We can also demonstrate this result geometrically: see Figure 15, for the case . Here, b² dots are arranged as eight triangles with an extra dot in the centre. Now

15. If b is odd, then b² has the form .

and so b² has the form .

In Chapter 1 we saw that the squares from 1² to 10² end with 1, 4, 5, 6, 9, or 0. Is this true for all squares? By the division rule, every integer can be written as , where , and so

So the square of ends with the same digit as the square of r, and so must also be 1, 4, 5, 6, 9 or 0. It follows that no perfect square can end in 2, 3, 7, or 8.

We can also use the division rule to obtain results involving cubes. A single example will give the idea.

Every cube has the form .

This is because every integer b has the form , or :

if , then ,

which has the form 9n with : for example, ;

if , then ,

which has the form with : for example, ;

if , then ,

which has the form : for example, .

So if then b³ has the form 9n, if then b³ has the form , and if then b³ has the form .

We conclude this section by stating without proof an intriguing result that links squares and cubes. In Chapter 1 we asked whether the sum of the first few positive cubes must always be a perfect square. But it can be proved that, for any number n,

and so the result is indeed true: for example,

Divisor tests

In our decimal counting system we can write any whole number, such as 47,972, as a sum of powers of 10 (where 10⁰ is taken to be 1):

and in general we can write any positive number in the form

Because our decimal system is a place-value system, we need to use only the ten digits

with the two 7s in 47,972 standing for 7000 and 70. We can therefore carry out our calculations in columns, with the columns representing units, tens, hundreds, thousands, …, as we move from right to left.

In a similar way, we can write any number as a sum of powers of 2 (the binary counting system used in computing), or of 12 (the duodecimal system used for feet and inches, and formerly in Britain for shillings and pence), or of any other integer greater than 1, and many statements about the decimal system have their analogues in these other systems too. For simplicity we’ll consider only decimal numbers here.

In this section we’ll see some tests that tell us very quickly whether a given positive number is divisible by the integers 2, 3, 4, 5, 8, 9, 10, 11, and 25, and we’ll also explain why they work. We’ll also show how to test for divisibility by some larger numbers.

As we saw in Chapter 1, we can easily test the number for divisibility by 10 and by 5 by checking its last digit, a₀:

Divisibility by 10: n is divisible by 10 if and only if its last digit is 0.

Divisibility by 5: n is divisible by 5 if and only if its last digit is 0 or 5.

For example, 19,720 is divisible by both 10 and 5, and 19,725 is divisible by 5.

These are because if

then 10 divides all the powers of 10, except for , and so n is divisible by 10 if and only if 10 divides a₀—that is, , and so n ends in 0. Likewise, 5 also divides all powers of 10, except for 10⁰, and so n is divisible by 5 if and only if 5 divides a₀—that is, or 5, and so n ends in 0 or 5.

We can likewise test for divisibility by 25 by checking its last two digits, a₁ and a₀:

Divisibility by 25: n is divisible by 25 if and only if its last two digits are 00, 25, 50, or 75.

For example, 19,725 is divisible by 25.

This is because 25 divides all the powers of 10, except for 10¹ and 10⁰, and so n is divisible by 25 if and only if 25 divides the two-digit number —that is, n ends in 00, 25, 50, or 75.

In Chapter 1 we saw that.

Divisibility by 2: n is divisible by 2 if and only if its last digit is 2, 4, 6, 8, or 0.

For example, 19,726 is divisible by 2.

This is because 2 divides all the powers of 10, except for 10⁰, and so n is divisible by 2 if and only if 2 divides a₀—that is, its last digit is even.

We can likewise test for divisibility by 4 or by 8 by checking the last two or