Can you find any perfect numbers? Read this article to find out more...

The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?

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. . . .

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

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?

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.

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" .

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

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?

Is there an efficient way to work out how many factors a large number has?

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. . . .

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

How many zeros are there at the end of the number which is the product of first hundred positive integers?

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.

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...

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Find the number which has 8 divisors, such that the product of the divisors is 331776.

Can you find what the last two digits of the number $4^{1999}$ are?

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. . . .

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 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.

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

Explore the relationship between simple linear functions and their graphs.

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

I'm thinking of a number. When my number is divided by 5 the remainder is 4. When my number is divided by 3 the remainder is 2. Can you find my number?

Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

Can you find a way to identify times tables after they have been shifted up?

The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?

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?

How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

Find the highest power of 11 that will divide into 1000! exactly.

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. . . .

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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!?

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?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

I put eggs into a basket in groups of 7 and noticed that I could easily have divided them into piles of 2, 3, 4, 5 or 6 and always have one left over. How many eggs were in the basket?

What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A

Have you seen this way of doing multiplication ?

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.