What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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.
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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...
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" .
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?
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?
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
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.
Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?
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. . . .
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
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). . . .
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?
Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Find the number which has 8 divisors, such that the product of the divisors is 331776.
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?
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.
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.
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Can you find any perfect numbers? Read this article to find out more...
Play this game and see if you can figure out the computer's chosen number.
The clues for this Sudoku are the product of the numbers in adjacent squares.
Is there an efficient way to work out how many factors a large number has?
The flow chart requires two numbers, M and N. Select several values for M and try to establish what the flow chart does.
How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?
How many zeros are there at the end of the number which is the product of first hundred positive integers?
The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?
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?
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...
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. . . .
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.
Given the products of adjacent cells, can you complete this Sudoku?
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...
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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.
Nine squares are fitted together to form a rectangle. Can you find its dimensions?
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?
Can you find any two-digit numbers that satisfy all of these statements?
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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. . . .
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
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?