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?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
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?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?
How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?
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!?
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 highest power of 11 that will divide into 1000! exactly.
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
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.
What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A
Find the number which has 8 divisors, such that the product of the divisors is 331776.
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Is there an efficient way to work out how many factors a large number has?
Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?
Play this game and see if you can figure out the computer's chosen number.
Can you find any perfect numbers? Read this article to find out more...
Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?
How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
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.
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...
Follow this recipe for sieving numbers and see what interesting patterns emerge.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Can you work out what size grid you need to read our secret message?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?
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.
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?
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.
Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .
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?
Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?
I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...
Given the products of adjacent cells, can you complete this Sudoku?
This article for teachers describes how number arrays can be a useful representation for many number concepts.
What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?
Number problems at primary level that may require resilience.
Your vessel, the Starship Diophantus, has become damaged in deep space. Can you use your knowledge of times tables and some lightning reflexes to survive?
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?
When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .
Can you find a way to identify times tables after they have been shifted up or down?
Can you work out how many lengths I swim each day?
Can you work out what a ziffle is on the planet Zargon?
56 406 is the product of two consecutive numbers. What are these two numbers?
Can you find any two-digit numbers that satisfy all of these statements?
Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings?