Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
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?
The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?
Can you select the missing digit(s) to find the largest multiple?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
How many zeros are there at the end of the number which is the product of first hundred positive integers?
Find the highest power of 11 that will divide into 1000! exactly.
What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?
What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A
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 number which has 8 divisors, such that the product of the divisors is 331776.
The flow chart requires two numbers, M and N. Select several values for M and try to establish what the flow chart does.
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
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!?
This article explains various divisibility rules and why they work. An article to read with pencil and paper handy.
How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?
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?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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.
The clues for this Sudoku are the product of the numbers in adjacent squares.
Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?
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...
Is there an efficient way to work out how many factors a large number has?
Given the products of adjacent 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?
Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?
How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
Can you make lines of Cuisenaire rods that differ by 1?
Nine squares are fitted together to form a rectangle. Can you find its dimensions?
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. . . .
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Can you work out what size grid you need to read our secret message?
Follow this recipe for sieving numbers and see what interesting patterns emerge.
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.
What is the smallest number of answers you need to reveal in order to work out the missing headers?
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?
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 power.
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.
How many different number families can you find?
In how many ways can the number 1 000 000 be expressed as the product of three positive integers?
Can you find any two-digit numbers that satisfy all of these statements?
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 work out how many lengths I swim each day?
How many noughts are at the end of these giant numbers?
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...