Find a condition which determines whether the hyperbola y^2 - x^2 =
k contains any points with integer coordinates.
Why are there only a few lattice points on a hyperbola and
infinitely many on a parabola?
Explore the factors of the numbers which are written as 10101 in
different number bases. Prove that the numbers 10201, 11011 and
10101 are composite in any base.
Find the maximum value of 1/p + 1/q + 1/r where this sum is less than 1 and p, q, and r are positive integers.
Explore the properties of some groups such as: The set of all real
numbers excluding -1 together with the operation x*y = xy + x + y.
Find the identity and the inverse of the element x.
Let N be a six digit number with distinct digits. Find the number N given that the numbers N, 2N, 3N, 4N, 5N, 6N, when written underneath each other, form a latin square (that is each row and each. . . .
Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two digit numbers are multiplied to give a four digit number, so that the expression is correct. How many different solutions can you find?
The symbol [ ] means 'the integer part of'. Can the numbers [2x];
2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three
Euler found four whole numbers such that the sum of any two of the
numbers is a perfect square. Three of the numbers that he found are
a = 18530, b=65570, c=45986. Find the fourth number, x. You. . . .
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?
Ranging from kindergarten mathematics to the fringe of research
this informal article paints the big picture of number in a non
technical way suitable for primary teachers and older students.
Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2
Take any whole number q. Calculate q^2 - 1. Factorize
q^2-1 to give two factors a and b (not necessarily q+1 and q-1). Put c = a + b + 2q . Then you will find that ab+1 , bc+1 and ca+1 are all. . . .
How can we solve equations like 13x + 29y = 42 or 2x +4y = 13 with
the solutions x and y being integers? Read this article to find
Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?
Write 100 as the sum of two positive integers, one divisible by 7 and the other divisible by 11.
Then find formulas giving all the solutions to
7x + 11y = 100
where x and y are integers.
Using the 8 dominoes make a square where each of the columns and rows adds up to 8
To make 11 kilograms of this blend of coffee costs £15 per
kilogram. The blend uses more Brazilian, Kenyan and Mocha coffee...
How many kilograms of each type of coffee are used?
Investigate the sequences obtained by starting with any positive 2
digit number (10a+b) and repeatedly using the rule 10a+b maps to
10b-a to get the next number in the sequence.
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Let a(n) be the number of ways of expressing the integer n as an
ordered sum of 1's and 2's. Let b(n) be the number of ways of
expressing n as an ordered sum of integers greater than 1. (i)
Calculate. . . .
I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of. . . .