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?
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
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?
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?
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.
What is the smallest number with exactly 14 divisors?
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?
The clues for this Sudoku are the product of the numbers in adjacent squares.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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.
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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.
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Got It game for an adult and child. How can you play so that you know you will always win?
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?
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. . . .
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
Given the products of adjacent cells, can you complete this Sudoku?
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. . . .
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?
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.
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the 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.
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 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?
A game that tests your understanding of remainders.
Can you find a way to identify times tables after they have been shifted up?
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...
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Given the products of diagonally opposite cells - can you complete this Sudoku?
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. . . .
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. . . .
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?"
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 largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
What is the value of the digit A in the sum below: [3(230 + A)]^2 =
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?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
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!?
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
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. . . .
Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Have you seen this way of doing multiplication ?