A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
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 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?
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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?
Can you convince me of each of the following: If a square number is
multiplied by a square number the product is ALWAYS a square
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. . . .
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
What is the smallest number with exactly 14 divisors?
Prove that if the integer n is divisible by 4 then it can be written as the difference of two 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. . . .
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
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.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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?
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
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.
Can you find any perfect numbers? Read this article to find out more...
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.
What is the value of the digit A in the sum below: [3(230 + A)]^2 =
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
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?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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?
Explore the relationship between simple linear functions and their
Which pairs of cogs let the coloured tooth touch every tooth on the
other cog? Which pairs do not let this happen? Why?
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.
Can you find a way to identify times tables after they have been shifted up?
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
Find the highest power of 11 that will divide into 1000! exactly.
Have you seen this way of doing multiplication ?
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.
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...
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
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. . . .
Given the products of diagonally opposite cells - can you complete this Sudoku?
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?"