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

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

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" .

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?

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 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?

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.

How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?

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?

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

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

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 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?

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?

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.

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

The clues for this Sudoku are the product of the numbers in adjacent squares.

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

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.

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

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!?

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

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?

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

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

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

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

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.

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

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

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

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

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

Explore the relationship between simple linear functions and their graphs.

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.

Have you seen this way of doing multiplication ?

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

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

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

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?

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

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.

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

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?

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. . . .