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?

This package contains a collection of problems from the NRICH website that could be suitable for students who have a good understanding of Factors and Multiples and who feel ready to take on some. . . .

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

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

Given the products of diagonally opposite cells - can you complete this Sudoku?

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?

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

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.

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?

Given the products of adjacent cells, can you complete this Sudoku?

In how many ways can the number 1 000 000 be expressed as the product of three positive integers?

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

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.

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

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

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. . . .

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 collection of resources to support work on Factors and Multiples at Secondary level.

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?

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

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

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

Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.

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 relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

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.

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.

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

Follow this recipe for sieving numbers and see what interesting patterns emerge.

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

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. . . .

A game that tests your understanding of remainders.

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.

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?"

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.

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?

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

Substitution and Transposition all in one! How fiendish can these codes get?

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

Can you work out what size grid you need to read our secret message?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

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?

Have you seen this way of doing multiplication ?

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. . . .

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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?

Make a line of green and a line of yellow rods so that the lines differ in length by one (a white rod)