What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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?
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.
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
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.
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?
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?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by
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.
What is the smallest number with exactly 14 divisors?
Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
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...
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 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 =
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
In this article for teachers, Bernard Bagnall describes how to find
digital roots and suggests that they can be worth exploring when
confronted by a sequence of numbers.
Take any four digit number. Move the first digit to the 'back of
the queue' and move the rest along. Now add your two numbers. What
properties do your answers always have?
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.
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
Find the highest power of 11 that will divide into 1000! exactly.
Look on the back of any modern book and you will find an ISBN code. Take this code and calculate this sum in the way shown. Can you see what the answers always have in common?
How many four digit square numbers are composed of even numerals?
What four digit square numbers can be reversed and become the
square of another number?
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. . . .
In a square in which the houses are evenly spaced, numbers 3 and 10
are opposite each other. What is the smallest and what is the
largest possible number of houses in the square?
The flow chart requires two numbers, M and N. Select several values
for M and try to establish what the flow chart does.
Think of any three-digit number. Repeat the digits. The 6-digit
number that you end up with is divisible by 91. Is this a
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!?
Amazing as it may seem the three fives remaining in the following
`skeleton' are sufficient to reconstruct the entire long division
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?