Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?
In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?
Find the number which has 8 divisors, such that the product of the divisors is 331776.
Can you find a way to identify times tables after they have been shifted up?
A game that tests your understanding of remainders.
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?
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?
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.
Can you find any perfect numbers? Read this article to find out more...
Find the highest power of 11 that will divide into 1000! exactly.
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .
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?
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" .
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?
What is the smallest number with exactly 14 divisors?
How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?
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. . . .
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 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?"
Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
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. . . .
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
A game in which players take it in turns to choose a number. Can you block your opponent?
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?
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?
What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A
Given the products of adjacent cells, can you complete this Sudoku?
A mathematician goes into a supermarket and buys four items. Using a calculator she multiplies the cost instead of adding them. How can her answer be the same as the total at the till?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Follow this recipe for sieving numbers and see what interesting patterns emerge.
What is the smallest number of answers you need to reveal in order to work out the missing headers?
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.
Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?
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?
How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?
You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the 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?
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?
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...
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.
Given the products of diagonally opposite cells - can you complete this Sudoku?
Can you find what the last two digits of the number $4^{1999}$ are?
A challenge that requires you to apply your knowledge of the properties of numbers. Can you fill all the squares on the board?
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
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.