How many noughts are at the end of these giant numbers?
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. . . .
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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 convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...
In how many ways can the number 1 000 000 be expressed as the product of three positive integers?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
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). . . .
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?
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. . . .
How many zeros are there at the end of the number which is the product of first hundred positive integers?
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
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.
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
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.
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?
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 I. . . .
Make a line of green and a line of yellow rods so that the lines differ in length by one (a white rod)
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. . . .
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?
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?
Find the highest power of 11 that will divide into 1000! exactly.
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!?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?
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?
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Can you find any perfect numbers? Read this article to find out more...
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?
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?
Can you find any two-digit numbers that satisfy all of these statements?
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.
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
Substitution and Transposition all in one! How fiendish can these codes get?
The clues for this Sudoku are the product of the numbers in adjacent squares.
Can you explain the strategy for winning this game with any target?
Follow this recipe for sieving numbers and see what interesting patterns emerge.
Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.
Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?