What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
How many zeros are there at the end of the number which is the
product of first hundred positive integers?
What is the value of the digit A in the sum below: [3(230 + A)]^2 =
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. . . .
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?
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
A collection of resources to support work on Factors and Multiples at Secondary level.
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
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.
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?
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?
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 puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.
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?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
Find the highest power of 11 that will divide into 1000! exactly.
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!?
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?
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
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.
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). . . .
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?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
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?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
A challenge that requires you to apply your knowledge of the
properties of numbers. Can you fill all the squares on the board?
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
In how many ways can the number 1 000 000 be expressed as the
product of three positive integers?
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. . . .
Can you find a way to identify times tables after they have been shifted up?
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?
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. . . .
The clues for this Sudoku are the product of the numbers in adjacent squares.
Which pairs of cogs let the coloured tooth touch every tooth on the
other cog? Which pairs do not let this happen? Why?
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
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. . . .
Follow this recipe for sieving numbers and see what interesting patterns emerge.
What is the smallest number of answers you need to reveal in order
to work out the missing headers?
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?