Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
A collection of resources to support work on Factors and Multiples at Secondary level.
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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. . . .
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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.
In how many ways can the number 1 000 000 be expressed as the
product of three positive integers?
115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?
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?
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 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 puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
How many zeros are there at the end of the number which is the
product of first hundred positive integers?
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). . . .
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
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
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.
Given the products of diagonally opposite cells - can you complete this Sudoku?
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?
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
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?
A game that tests your understanding of remainders.
The clues for this Sudoku are the product of the numbers in adjacent squares.
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?
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?
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
What is the value of the digit A in the sum below: [3(230 + A)]^2 =
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
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?
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!?
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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. . . .
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
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?
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.
Can you find any perfect numbers? Read this article to find out more...
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
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.
Can you find a way to identify times tables after they have been shifted up?