This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
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 =
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
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?
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. . . .
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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?
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?
Substitution and Transposition all in one! How fiendish can these codes get?
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
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.
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
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?
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). . . .
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.
A collection of resources to support work on Factors and Multiples at Secondary level.
Find the highest power of 11 that will divide into 1000! exactly.
Can you work out what size grid you need to read our secret message?
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
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 the number which has 8 divisors, such that the product of the
divisors is 331776.
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
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.
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?
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
In this activity, the computer chooses a times table and shifts it.
Can you work out the table and the shift each time?
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
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. . . .
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Can you find a way to identify times tables after they have been shifted up?
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?
A game that tests your understanding of remainders.
The clues for this Sudoku are the product of the numbers in
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?
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?
Given the products of diagonally opposite cells - can you complete
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!?
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?
A challenge that requires you to apply your knowledge of the
properties of numbers. Can you fill all the squares on the board?