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?
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.
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?
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.
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?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
The clues for this Sudoku are the product of the numbers in adjacent squares.
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?
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
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. . . .
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?
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?
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. . . .
Given the products of adjacent cells, can you complete this Sudoku?
What is the smallest number with exactly 14 divisors?
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 puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Can you find a way to identify times tables after they have been shifted up?
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
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
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.
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
Find the highest power of 11 that will divide into 1000! exactly.
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?
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.
Given the products of diagonally opposite cells - can you complete this Sudoku?
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 =
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?"
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
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. . . .
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?
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
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 largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
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. . . .
Have you seen this way of doing multiplication ?
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
A game in which players take it in turns to choose a number. Can you block your opponent?
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
Can you work out what size grid you need to read our secret message?