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.
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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Can you find any perfect numbers? Read this article to find out more...
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?
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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. . . .
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?
A student in a maths class was trying to get some information from
her teacher. She was given some clues and then the teacher ended by
saying, "Well, how old are they?"
Given the products of adjacent cells, can you complete this Sudoku?
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.
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
When the number x 1 x x x is multiplied by 417 this gives the
answer 9 x x x 0 5 7. Find the missing digits, each of which is
represented by an "x" .
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?
A game that tests your understanding of remainders.
Given the products of diagonally opposite cells - can you complete this Sudoku?
Is there an efficient way to work out how many factors a large number has?
How many zeros are there at the end of the number which is the
product of first hundred positive integers?
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?
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.
Which pairs of cogs let the coloured tooth touch every tooth on the
other cog? Which pairs do not let this happen? Why?
Follow this recipe for sieving numbers and see what interesting patterns emerge.
Find the frequency distribution for ordinary English, and use it to help you crack the code.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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. . . .
Can you find a way to identify times tables after they have been shifted up?
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 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?
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.
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
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
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. . . .
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
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?
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
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?
Complete the following expressions so that each one gives a four
digit number as the product of two two digit numbers and uses the
digits 1 to 8 once and only once.
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
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.