How many right-angled triangles with whole-number sides have a side of length 29?
Triples of numbers (a, b, c) for which a, b, and c are whole numbers and are called Pythagorean triples, and correspond to side-lengths of right-angled triangles. As we discovered in Chapter 5, there are only two such Pythagorean triples in which one side has length 29: these are (20, 21, 29) and (29, 420, 421).
Are any of the numbers 11, 111, 1111, 11111, . . . perfect squares?
In Chapter 2 we showed that all perfect squares must have the form 4n or , for some integer n. The numbers in this question all have the form , and so none of them can be a perfect square.
I have some eggs. When arranged in rows of 3 there are 2 left over, in rows of 5 there are 3 left over, and in rows of 7 there are 2 left over. How many eggs are there altogether?
In Chapter 4 we discussed the solution of simultaneous linear congruences, and gave this problem as an example. It is a version of Sunzi’s problem, and asks for the solution of the simultaneous congruences , , and . The smallest solution is , and other answers are found by adding multiples of .
Can one construct a regular polygon with 100 sides if measuring is forbidden?
In Chapter 3 we noted a connection between the construction of regular polygons and the Fermat primes mentioned above: this is Gauss’s result that a regular polygon with n sides can be constructed by an unmarked ruler and compasses if and only if n is the product of a power of 2 and unequal Fermat primes. But , where the Fermat prime 5 is repeated, and so a regular polygon with 100 sides cannot be so constructed.
How many shuffles are needed to restore the order of the cards in a pack with two Jokers?
Questions involving the shuffling of cards can be answered with help from Fermat’s little theorem on primes, as we described in Chapter 6. In this case, where there are 54 cards, the answer is 20 shuffles.
If I can buy partridges for 3 cents, pigeons for 2 cents, and two sparrows for a cent, and if I spend 30 pence on buying 30 birds, how many birds of each kind must I buy?
This is a Diophantine problem requiring an integer solution. As we discussed in Chapter 5, the solution involves two linear equations in three unknowns, together with the extra requirement that the number of each kind of bird that I buy must be a positive integer. The only answer is 5 partridges, 3 pigeons, and 22 sparrows.
How do prime numbers help to keep our credit cards secure?
In Chapter 6 we saw that multiplying large primes is usually a simple matter, whereas trying to factorize large numbers into their prime factors is not. The RSA method of encryption and decryption recognizes this difficulty and involves Euler’s theorem, a generalization of Fermat’s little theorem.
What is the Riemann hypothesis, and how can I earn a million dollars?
The Riemann hypothesis (or conjecture) was discussed in Chapter 8, and is one of the most famous unsolved problems in mathematics. The conjecture is closely related to that of locating the places where a certain function (called the ‘zeta function’) has a zero value. It turns out that its proof would tell us much about how prime numbers are distributed, as well as providing its solver with lasting fame and an award of one million dollars by the Clay Mathematics Institute.
Integers
How can we recognize whether a given number, such as 12,345,678, is a multiple of 8? or of 9? or of 11? or of 88?
In Chapter 2 we discussed several tests for divisibility by various numbers, such as 8, 9, and 11. The given number is not a multiple of 8 because the number given by its last three digits (678) is not divisible by 8. The sum of its digits is 36, which is divisible by 9, and so the given number is divisible by 9. The alternating sum of its digits is −4, which is not divisible by 11, and so the given number is not divisible by 11 or by any of its multiples, such as 88.
Squares and cubes
We discussed various topics related to squares and higher powers in Chapters 2 and 5. The next three questions were answered in the section on Squares in Chapter 2.
Do any squares end in 2, 3, 7, or 8?
We showed that all squares must end in 0, 1, 4, 5, 6, or 9, and so there can be no squares that end in 2, 3, 7, or 8.
Must all squares be of the form 4n or , where n is an integer?
As mentioned above, we showed that all squares must be of these forms—in particular, squares of even numbers all have the form 4n, and squares of odd numbers all have the form . We can also express this by saying that all squares must be congruent to 0 or 1 (mod 4).
Must the sum of the first few odd numbers, 1, 3, 5, 7, …, always be a square?
For any number k, the first k odd numbers are 1, 3, 5, …, , and on adding these numbers together we find that their sum is k2. So the sum of the first few odd numbers is always a square.
The next two questions were discussed in Chapter 5, where we investigated which numbers can be written as the sum of two or more squares.
Which numbers can be written as the sum of two squares?
Because every square is congruent to 0 or 1 (mod 4), the sum of two squares must be congruent to 0, 1, or 2
