This is part of our collection of Short Problems.
You may also be interested in our longer problems on Factors, Multiples and Primes.
Printable worksheets containing selections of these problems are available here.
You may also be interested in our longer problems on Factors, Multiples and Primes.
Printable worksheets containing selections of these problems are available here.
problem
What's on the back?
Age
11 to 14
Challenge level
Four cards have a number on one side and a phrase on the back. On each card, the number does not have the property described on the back. What do the four cards have on them?
problem
Product 100
Age
11 to 14
Challenge level
The product of four different positive integers is 100. What is the sum of these four integers?
problem
Factor trio
Age
11 to 14
Challenge level
Which of the numbers shown is the product of exactly 3 distinct prime factors?
problem
Reversible primes
Age
11 to 14
Challenge level
How many two-digit primes are there between 10 and 99 which are also prime when reversed?
problem
Calculation 2000
Age
11 to 14
Challenge level
Weekly Problem 49 - 2013
What is the value of 2000 + 1999 × 2000?
problem
Prime order
Age
11 to 14
Challenge level
Weekly Problem 24 - 2006
How many of the rearrangements of the digits 1, 3 and 5 give prime numbers?
How many of the rearrangements of the digits 1, 3 and 5 give prime numbers?
problem
Almost a million
Age
11 to 14
Challenge level
Weekly Problem 26 - 2014
Which of the given numbers is divisible by 6?
Which of the given numbers is divisible by 6?
problem
Divisible digits
Age
11 to 14
Challenge level
Can you find the missing digits, given that the number is divisible by 3, 4, 5 and 6?
problem
Tricky customer
Age
11 to 14
Challenge level
Charlie doesn't want his new house number to be divisible by 3 or 5. How many choices of house does he have?
problem
Producing zeros
Age
11 to 14
Challenge level
Weekly Problem 10 - 2008
If the numbers 1 to 10 are all multiplied together, how many zeros are at the end of the answer?
If the numbers 1 to 10 are all multiplied together, how many zeros are at the end of the answer?
problem
Multiple years
Age
11 to 14
Challenge level
Weekly Problem 18 - 2016
The year 2010 is one in which the sum of the digits is a factor of the year itself. What is the next year that has the same property?
The year 2010 is one in which the sum of the digits is a factor of the year itself. What is the next year that has the same property?
problem
Pairing up
Age
11 to 14
Challenge level
The numbers 72, 8, 24, 10, 5, 45, 36, 15 are grouped in pairs so that each pair has the same product. Which number is paired with 10?
problem
Stamp collecting
Age
11 to 14
Challenge level
Last week, Tom and Sophie bought some stamps and they spent exactly £10. Can you work out how many stamps they bought?
problem
Little Goldbach
Age
11 to 14
Challenge level
How many of the numbers 1 to 20 are not the sum of two primes?
problem
Multiplication table puzzle
Age
11 to 14
Challenge level
In the multiplication table on the right, only some of the numbers have been given. What is the value of A+B+C+D+E?
problem
AB search
Age
11 to 14
Challenge level
The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?
problem
Adjacent factors
Age
11 to 14
Challenge level
Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?
problem
Back of the queue
Age
11 to 14
Challenge level
Weekly Problem 48 - 2013
What is the remainder when the number 743589 × 301647 is divided by 5?
problem
Digital division
Age
11 to 14
Challenge level
How many three digit numbers formed with three different digits from 0, 1, 2, 3 and 5 are divisible by 6?
problem
One short
Age
11 to 14
Challenge level
A whole number less than 100 gives remainders of 2, 3 and 4 when divided by 3, 4 and 5. What is the remainder when it is divided by 7?
problem
Red card blue card
Age
11 to 14
Challenge level
Can you arrange the red and blue cards so that the rules are all followed?
problem
Cakes and buns
Age
11 to 14
Challenge level
Helen buys some cakes and some buns for her party. Can you work out how many of each she buys?
problem
Find from factors
Age
11 to 14
Challenge level
Weekly Problem 35 - 2006
A number has exactly eight factors, two of which are 21 and 35. What is the number?
A number has exactly eight factors, two of which are 21 and 35. What is the number?
problem
Halloween day
Age
11 to 14
Challenge level
One year there were exactly four Tuesdays and four Fridays in October. On what day of the week was Halloween.
problem
Grandma's cake
Age
11 to 14
Challenge level
What is the smallest number of pieces grandma should cut her cake into to guarantee each grandchild gets the same amount of cake and none is left over?
problem
Jenny's logic
Age
11 to 14
Challenge level
Weekly Problem 52 - 2009
How did Jenny figure out that Tom's cards added to an even number?
How did Jenny figure out that Tom's cards added to an even number?
problem
Smallest abundant number
Age
11 to 14
Challenge level
Weekly Problem 34 - 2017
An abundant number is a positive integer N such that the sum of the factors of N is larger than 2N. What is the smallest abundant number?
An abundant number is a positive integer N such that the sum of the factors of N is larger than 2N. What is the smallest abundant number?
problem
Ones, twos and threes
Age
11 to 14
Challenge level
Each digit of a positive integer is 1, 2 or 3, and each of these occurs at least twice. What is the smallest such integer that is not divisible by 2 or 3?
problem
Common remainder
Age
11 to 14
Challenge level
144 divided by n leaves a remainder of 11. 220 divided by n also leaves a remainder of 11. What is n?
problem
HCF expression
Age
11 to 14
Challenge level
Find out which two distinct primes less than $7$ will give the largest highest common factor of these two expressions.
problem
Leap monday
Age
11 to 14
Challenge level
Can you find the next time that the 29th of February will fall on a Monday?
problem
Four or five
Age
11 to 14
Challenge level
The diagram shows a large rectangle composed of 9 smaller rectangles. If each of these rectangles has integer sides, what could the area of the large rectangle be?
problem
Seven from nine
Age
11 to 14
Challenge level
In how many ways can seven of the numbers 1-9 be chosen such that they add up to a multiple of 3?
problem
Big blackboard
Age
11 to 14
Challenge level
Can you work out which numbers between 1 and 2016 have exactly two of 2, 3, 4 as factors?
problem
Colossal sum
Age
11 to 14
Challenge level
What is the units digit in this sum of powers of 9?
problem
Factor sum
Age
11 to 14
Challenge level
Given any positive integer n, Paul adds together the factors of n, apart from n itself. Which of the numbers 1, 3, 5, 7 and 9 can never be Paul's answer?
problem
Missing digit
Age
11 to 14
Challenge level
What digit must replace the star to make the number a multiple of 11?
problem
Cinema costs
Age
11 to 14
Challenge level
Weekly Problem 41 - 2009
At a cinema a child's ticket costs £4.20 and an adult's ticket costs £7.70. How much did it cost this group of adults and children to see a film?
problem
Powerful finale
Age
11 to 14
Challenge level
Weekly Problem 24 - 2017
What is the last digit of $3^{2011}$?
What is the last digit of $3^{2011}$?
problem
Indivisible
Age
14 to 16
Challenge level
Each time a class lines up in different sized groups, a different number of people are left over. How large can the class be?
problem
Essential supplies
Age
14 to 16
Challenge level
Chocolate bars come in boxes of 5 or boxes of 12. How many boxes do you need to have exactly 2005 chocolate bars?
problem
Punky fish
Age
14 to 16
Challenge level
A male punky fish has 9 stripes and a female punky fish has 8 stripes. I count 86 stripes on the fish in my tank. What is the ratio of male fish to female fish?
problem
End of a prime
Age
14 to 16
Challenge level
I made a list of every number that is the units digit of at least one prime number. How many digits appear in the list?
problem
Threes and fours
Age
14 to 16
Challenge level
What is the smallest integer where every digit is a 3 or a 4 and it is divisible by both 3 and 4?
problem
Peter's primes
Age
14 to 16
Challenge level
Peter wrote a list of all the numbers that can be formed by changing one digit of the number 200. How many of Peter's numbers are prime?
problem
Flora the florist
Age
14 to 16
Challenge level
Flora has roses in three colours. What is the greatest number of identical bunches she can make, using all the flowers?
problem
Long list
Age
14 to 16
Challenge level
Weekly Problem 47 - 2017
How many numbers do I need in a list to have two squares, two primes and two cubes?
How many numbers do I need in a list to have two squares, two primes and two cubes?
problem
Sticky fingers
Age
14 to 16
Challenge level
Ruth wants to puts stickers on the cuboid she has made from little cubes. Will she have any stickers left over?
problem
Times and square
Age
14 to 16
Challenge level
Roger multiplies two consecutive integers and squares the result. Can you find the last two digits of his new number?
problem
Eight factors only
Age
14 to 16
Challenge level
We are given two factors of a number with eight factors. Can you work out the other factors of the number?
problem
Divisible palindrome
Age
14 to 16
Challenge level
What is the sum of the digits of the largest 4-digit palindromic number which is divisible by 15?
problem
Relative time
Age
14 to 16
Challenge level
Albert Einstein is experimenting with two unusual clocks. At what time do they next agree?
problem
Powerful zeros
Age
14 to 16
Challenge level
How many zeros are there at the end of $3^4 \times 4^5 \times 5^6$?
problem
Coin collection
Age
14 to 16
Challenge level
When coins are put into piles of six 3 remain and in piles of eight 7 remain. How many remain when they are put into piles of 24?
problem
Added power
Age
14 to 16
Challenge level
How many integers $n$ are there for which $n$ and $n^3+3$ are both prime?
problem
Square sum
Age
14 to 16
Challenge level
One of these numbers is the largest of nine consecutive positive integers whose sum is a perfect square. Which one is it?
problem
Triangular algebra
Age
14 to 16
Challenge level
Weekly Problem 26 - 2017
The angles in the triangle are shown in the diagram in terms of x and y. If x and y are positive integers, what is the value of x+y?
The angles in the triangle are shown in the diagram in terms of x and y. If x and y are positive integers, what is the value of x+y?
problem
Factorised factorial
Age
14 to 16
Challenge level
Weekly Problem 17 - 2010
The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?
The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?
problem
Factor list
Age
14 to 16
Challenge level
Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?
problem
Adding a square to a cube
Age
14 to 16
Challenge level
If you take a number and add its square to its cube, how often will you get a perfect square?
problem
Rational integer
Age
14 to 16
Challenge level
Weekly Problem 39 - 2012
For how many values of $n$ are both $n$ and $\frac{n+3}{n-1}$ integers?
problem
Square product
Age
14 to 16
Challenge level
Weekly Problem 10 - 2011
Will this product give a perfect square?
Will this product give a perfect square?
problem
Cancelling fractions
Age
14 to 16
Challenge level
Can you find a fraction with the following properties?
problem
Fortunate inflation
Age
14 to 16
Challenge level
The price of an item in pounds and pence is increased by 4%. The new price is an integer number of pounds. Can you find it?
problem
Super computer
Age
14 to 16
Challenge level
What is the units digit of the given expression?
problem
Primes and six
Age
14 to 16
Challenge level
Weekly Problem 1 - 2015
If $p$ and $q$ are prime numbers greater than $3$ and $q=p+2$, prove that $pq+1$ is divisible by $36$.
If $p$ and $q$ are prime numbers greater than $3$ and $q=p+2$, prove that $pq+1$ is divisible by $36$.
problem
Integer indices
Age
14 to 16
Challenge level
From this sum of powers, can you find the sum of the indices?
problem
Leftovers
Age
14 to 16
Challenge level
Weekly Problem 26 - 2008
If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?
If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?
problem
Three primes
Age
14 to 16
Challenge level
Weekly Problem 6 - 2010
Can you find three primes such that their product is exactly five times their sum? Do you think you have found all possibilities?
Can you find three primes such that their product is exactly five times their sum? Do you think you have found all possibilities?
problem
Square LCM
Age
14 to 16
Challenge level
Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?