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 smallest number with exactly 14 divisors?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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?
A game that tests your understanding of remainders.
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...
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?
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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.
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
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?
Amazing as it may seem the three fives remaining in the following
`skeleton' are sufficient to reconstruct the entire long division
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
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!?
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?
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?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
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
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
What is the value of the digit A in the sum below: [3(230 + A)]^2 =
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 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 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?
Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
The flow chart requires two numbers, M and N. Select several values
for M and try to establish what the flow chart does.
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
Find the highest power of 11 that will divide into 1000! exactly.
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.
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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. . . .
You have worked out a secret code with a friend. Every letter in the alphabet can be represented by a binary value.
Show that if three prime numbers, all greater than 3, form an
arithmetic progression then the common difference is divisible by
6. What if one of the terms is 3?
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. . . .
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.
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?
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
The four digits 5, 6, 7 and 8 are put at random in the spaces of
the number : 3 _ 1 _ 4 _ 0 _ 9 2 Calculate the probability that the
answer will be a multiple of 396.
Lyndon chose this as one of his favourite problems. It is
accessible but needs some careful analysis of what is included and
what is not. A systematic approach is really helpful.
Suppose an operator types a US Bank check code into a machine and transposes two adjacent digits will the machine pick up every error of this type? Does the same apply to ISBN numbers; will a machine. . . .
Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check. . . .
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). . . .
Take any pair of numbers, say 9 and 14. Take the larger number,
fourteen, and count up in 14s. Then divide each of those values by
the 9, and look at the remainders.