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). . . .
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.
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
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.
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
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?
In how many ways can the number 1 000 000 be expressed as the
product of three positive integers?
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
Substitution and Transposition all in one! How fiendish can these codes get?
A collection of resources to support work on Factors and Multiples at Secondary level.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Given the products of diagonally opposite cells - can you complete this Sudoku?
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
Can you work out what size grid you need to read our secret message?
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.
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?
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
How many zeros are there at the end of the number which is the
product of first hundred positive integers?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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 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 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 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?
What is the value of the digit A in the sum below: [3(230 + A)]^2 =
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?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Have you seen this way of doing multiplication ?
Can you find any perfect numbers? Read this article to find out more...
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 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
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.
How many noughts are at the end of these giant numbers?
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. . . .
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
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 convince me of each of the following: If a square number is
multiplied by a square number the product is ALWAYS a square
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
Explore the relationship between simple linear functions and their
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?
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?
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Find the highest power of 11 that will divide into 1000! exactly.