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.
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" .
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?
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. . . .
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.
The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?
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 number...
What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
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). . . .
Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Is there an efficient way to work out how many factors a large number has?
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?
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?
How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?
Given the products of adjacent cells, can you complete this Sudoku?
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 three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
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?
How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?
Can you find any perfect numbers? Read this article to find out more...
Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?
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?
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.
Follow this recipe for sieving numbers and see what interesting patterns emerge.
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 largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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.
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 number which has 8 divisors, such that the product of the divisors is 331776.
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 largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
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?
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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...
The clues for this Sudoku are the product of the numbers in adjacent squares.
How many noughts are at the end of these giant numbers?
Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?
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.
Find the highest power of 11 that will divide into 1000! exactly.