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. . . .
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
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?
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?
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
Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). . . .
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
Explore the factors of the numbers which are written as 10101 in
different number bases. Prove that the numbers 10201, 11011 and
10101 are composite in any base.
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
Follow this recipe for sieving numbers and see what interesting patterns emerge.
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" .
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
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?
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
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. . . .
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
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?
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 work out what size grid you need to read our secret message?
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?
In how many ways can the number 1 000 000 be expressed as the
product of three positive integers?
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?
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
Can you find any perfect numbers? Read this article to find out more...
Complete the following expressions so that each one gives a four
digit number as the product of two two digit numbers and uses the
digits 1 to 8 once and only once.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
How many noughts are at the end of these giant numbers?
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.
Which pairs of cogs let the coloured tooth touch every tooth on the
other cog? Which pairs do not let this happen? Why?
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Have you seen this way of doing multiplication ?
Find the highest power of 11 that will divide into 1000! exactly.
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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 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?
Given the products of diagonally opposite cells - can you complete this Sudoku?
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!?
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.