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.
In how many ways can the number 1 000 000 be expressed as the
product of three positive integers?
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
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.
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?
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
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?
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
Given the products of adjacent cells, can you complete this Sudoku?
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides
exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest
power of two that divides exactly into 100!?
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. . . .
Find the highest power of 11 that will divide into 1000! exactly.
What is the smallest number of answers you need to reveal in order
to work out the missing headers?
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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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.
The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
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?
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?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Find the frequency distribution for ordinary English, and use it to help you crack the code.
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?
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 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?
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?
Can you work out what size grid you need to read our secret message?
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?
What is the smallest number with exactly 14 divisors?
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.
Explore the relationship between simple linear functions and their
Can you find a way to identify times tables after they have been shifted up?
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
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.
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.
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.
A game that tests your understanding of remainders.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Can you find any perfect numbers? Read this article to find out more...
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. . . .
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). . . .
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?