The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
Find the number which has 8 divisors, such that the product of the divisors is 331776.
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. . . .
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
How many zeros are there at the end of the number which is the product of first hundred positive integers?
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
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" .
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?
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.
Given the products of adjacent cells, can you complete this Sudoku?
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?
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...
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?
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.
Can you make lines of Cuisenaire rods that differ by 1?
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.
The clues for this Sudoku are the product of the numbers in adjacent squares.
Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Find the highest power of 11 that will divide into 1000! exactly.
Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?
Is there an efficient way to work out how many factors a large number has?
Can you find any perfect numbers? Read this article to find out more...
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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.
Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.
The items in the shopping basket add and multiply to give the same amount. What could their prices be?
The flow chart requires two numbers, M and N. Select several values for M and try to establish what the flow chart does.
What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?
The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?
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.
How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?
Can you find any two-digit numbers that satisfy all of these statements?
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
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 power.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
Can you work out what size grid you need to read our secret message?
Follow this recipe for sieving numbers and see what interesting patterns emerge.
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.
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 value of the digit A in the sum below: [3(230 + A)]^2 = 49280A
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?
What is the smallest number of answers you need to reveal in order to work out the missing headers?
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?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"