Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
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.
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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.
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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?
What is the smallest number with exactly 14 divisors?
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. . . .
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
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.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
The clues for this Sudoku are the product of the numbers in adjacent squares.
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.
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
I'm thinking of a number. When my number is divided by 5 the
remainder is 4. When my number is divided by 3 the remainder is 2.
Can you find my number?
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
Can you find a way to identify times tables after they have been shifted up?
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
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 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?
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
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.
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?
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...
Find the highest power of 11 that will divide into 1000! exactly.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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?
Given the products of diagonally opposite cells - can you complete this Sudoku?
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
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?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
What is the value of the digit A in the sum below: [3(230 + A)]^2 =
Some 4 digit numbers can be written as the product of a 3 digit
number and a 2 digit number using the digits 1 to 9 each once and
only once. The number 4396 can be written as just such a product.
Can. . . .
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!?
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?
Given the products of adjacent cells, can you complete this Sudoku?
What is the smallest number of answers you need to reveal in order
to work out the missing headers?
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.
Can you find any perfect numbers? Read this article to find out more...
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
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. . . .
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. . . .
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
A game that tests your understanding of remainders.
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)