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)
Can you work out what size grid you need to read our secret message?
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
A collection of resources to support work on Factors and Multiples at Secondary level.
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. . . .
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
In how many ways can the number 1 000 000 be expressed as the
product of three positive integers?
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?
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?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Substitution and Transposition all in one! How fiendish can these codes get?
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
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). . . .
Given the products of diagonally opposite cells - can you complete this Sudoku?
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?
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.
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
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.
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.
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?
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.
Can you find any perfect numbers? Read this article to find out more...
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
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?
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. . . .
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
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?
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?
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?
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
What is the smallest number of answers you need to reveal in order
to work out the missing headers?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
A number N is divisible by 10, 90, 98 and 882 but it is NOT
divisible by 50 or 270 or 686 or 1764. It is also known that N is a
factor of 9261000. What is N?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
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
Explore the relationship between simple linear functions and their
Find the highest power of 11 that will divide into 1000! exactly.
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.