Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
Make a set of numbers that use all the digits from 1 to 9, once and
once only. Add them up. The result is divisible by 9. Add each of
the digits in the new number. What is their sum? Now try some. . . .
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I. . . .
115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Can you convince me of each of the following: If a square number is
multiplied by a square number the product is ALWAYS a square
A mathematician goes into a supermarket and buys four items. Using
a calculator she multiplies the cost instead of adding them. How
can her answer be the same as the total at the till?
Is there an efficient way to work out how many factors a large number has?
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.
The clues for this Sudoku are the product of the numbers in adjacent squares.
Can you find any perfect numbers? Read this article to find out more...
Ben passed a third of his counters to Jack, Jack passed a quarter
of his counters to Emma and Emma passed a fifth of her counters to
Ben. After this they all had the same number of counters.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
The nth term of a sequence is given by the formula n^3 + 11n . Find
the first four terms of the sequence given by this formula and the
first term of the sequence which is bigger than one million. . . .
Find the smallest positive integer N such that N/2 is a perfect
cube, N/3 is a perfect fifth power and N/5 is a perfect seventh
How many noughts are at the end of these giant numbers?
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
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 five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
How many zeros are there at the end of the number which is the
product of first hundred positive integers?
Data is sent in chunks of two different sizes - a yellow chunk has
5 characters and a blue chunk has 9 characters. A data slot of size
31 cannot be exactly filled with a combination of yellow and. . . .
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
Substitution and Transposition all in one! How fiendish can these codes get?
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
Given the products of diagonally opposite cells - can you complete this Sudoku?
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?
Can you work out what size grid you need to read our secret message?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
Which pairs of cogs let the coloured tooth touch every tooth on the
other cog? Which pairs do not let this happen? Why?
Can you find a relationship between the number of dots on the
circle and the number of steps that will ensure that all points are
Follow this recipe for sieving numbers and see what interesting patterns emerge.