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?
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. . . .
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 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.
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. . . .
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
Got It game for an adult and child. How can you play so that you know you will always win?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
What is the smallest number with exactly 14 divisors?
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.
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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 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.
The clues for this Sudoku are the product of the numbers in adjacent squares.
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
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?
Can you work out what size grid you need to read our secret message?
Substitution and Transposition all in one! How fiendish can these codes get?
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?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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 the frequency distribution for ordinary English, and use it to help you crack the code.
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
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
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. . . .
Can you find a way to identify times tables after they have been shifted up?
A game that tests your understanding of remainders.
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
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Can you find any perfect numbers? Read this article to find out more...
Follow this recipe for sieving numbers and see what interesting patterns emerge.
Find the highest power of 11 that will divide into 1000! exactly.
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
Given the products of diagonally opposite cells - can you complete this Sudoku?
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?
What is the value of the digit A in the sum below: [3(230 + A)]^2 =
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.