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. . . .
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
Can you work out what size grid you need to read our secret message?
How many zeros are there at the end of the number which is the
product of first hundred positive integers?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
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.
In how many ways can the number 1 000 000 be expressed as the
product of three positive integers?
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
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?
Substitution and Transposition all in one! How fiendish can these codes get?
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
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?
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.
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
Given the products of diagonally opposite cells - can you complete this Sudoku?
Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). . . .
A collection of resources to support work on Factors and Multiples at Secondary level.
Three people chose this as a favourite problem. It is the sort of
problem that needs thinking time - but once the connection is made
it gives access to many similar ideas.
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
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?
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?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
Take any pair of numbers, say 9 and 14. Take the larger number,
fourteen, and count up in 14s. Then divide each of those values by
the 9, and look at the remainders.
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. . . .
Can you find any perfect numbers? Read this article to find out more...
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
I put eggs into a basket in groups of 7 and noticed that I could
easily have divided them into piles of 2, 3, 4, 5 or 6 and always
have one left over. How many eggs were in the basket?
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
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?
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?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
The number 8888...88M9999...99 is divisible by 7 and it starts with
the digit 8 repeated 50 times and ends with the digit 9 repeated 50
times. What is the value of the digit M?
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
What is the smallest number of answers you need to reveal in order
to work out the missing headers?
Follow this recipe for sieving numbers and see what interesting patterns emerge.
Which pairs of cogs let the coloured tooth touch every tooth on the
other cog? Which pairs do not let this happen? 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. . . .
Find the highest power of 11 that will divide into 1000! exactly.
Given the products of adjacent cells, can you complete this Sudoku?
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?