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. . . .
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
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...
What is the smallest number with exactly 14 divisors?
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 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.
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 a large number of 1s, 4s, 7s and 10s. What numbers can we make?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
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?
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.
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. . . .
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?"
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?
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.
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
Given the products of adjacent cells, can you complete this Sudoku?
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. . . .
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. . . .
Can you find a way to identify times tables after they have been shifted up?
A game that tests your understanding of remainders.
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?
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 clues for this Sudoku are the product of the numbers in adjacent squares.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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
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?
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?
A game in which players take it in turns to choose a number. Can you block your opponent?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
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 =
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
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
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?
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 smallest number of answers you need to reveal in order
to work out the missing headers?
Can you work out what size grid you need to read our secret message?
Find the highest power of 11 that will divide into 1000! exactly.
Follow this recipe for sieving numbers and see what interesting patterns emerge.
Can you find any perfect numbers? Read this article to find out more...
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?