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 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?
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?
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.
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 three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Four of these clues are needed to find the chosen number on this
grid and four are true but do nothing to help in finding the
number. Can you sort out the clues and find the number?
On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?
Is there an efficient way to work out how many factors a large number has?
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
What is the value of the digit A in the sum below: [3(230 + A)]^2 =
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?
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Number problems at primary level that may require determination.
Have a go at balancing this equation. Can you find different ways of doing it?
Can you work out some different ways to balance this equation?
Find the highest power of 11 that will divide into 1000! exactly.
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?
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. . . .
Got It game for an adult and child. How can you play so that you know you will always win?
48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?
What happens if you join every second point on this circle? How
about every third point? Try with different steps and see if you
can predict what will happen.
What is the lowest number which always leaves a remainder of 1 when
divided by each of the numbers from 2 to 10?
Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?
Can you find any perfect numbers? Read this article to find out more...
Can you find a way to identify times tables after they have been shifted up?
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?
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.
In this problem we are looking at sets of parallel sticks that
cross each other. What is the least number of crossings you can
make? And the greatest?
Number problems at primary level to work on with others.
I throw three dice and get 5, 3 and 2. Add the scores on the three
dice. What do you get? Now multiply the scores. What do you notice?
56 406 is the product of two consecutive numbers. What are these
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
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!?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
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?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
The clues for this Sudoku are the product of the numbers in adjacent squares.
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.
This article for teachers describes how number arrays can be a
useful reprentation for many number concepts.
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 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?
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. . . .
Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?