List

Factors, Multiple and Primes - Short Problems

What's on the Back?
problem

What's on the back?

Age
11 to 14
Challenge level
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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?
Product 100
problem

Product 100

Age
11 to 14
Challenge level
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The product of four different positive integers is 100. What is the sum of these four integers?
Factor Trio
problem

Factor trio

Age
11 to 14
Challenge level
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Which of the numbers shown is the product of exactly 3 distinct prime factors?
Reversible Primes
problem

Reversible primes

Age
11 to 14
Challenge level
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How many two-digit primes are there between 10 and 99 which are also prime when reversed?
Calculation 2000
problem

Calculation 2000

Age
11 to 14
Challenge level
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Weekly Problem 49 - 2013
What is the value of 2000 + 1999 × 2000?
Prime Order
problem

Prime order

Age
11 to 14
Challenge level
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Weekly Problem 24 - 2006
How many of the rearrangements of the digits 1, 3 and 5 give prime numbers?
Almost a Million
problem

Almost a million

Age
11 to 14
Challenge level
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Weekly Problem 26 - 2014
Which of the given numbers is divisible by 6?
Divisible digits
problem

Divisible digits

Age
11 to 14
Challenge level
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Can you find the missing digits, given that the number is divisible by 3, 4, 5 and 6?
Tricky Customer
problem

Tricky customer

Age
11 to 14
Challenge level
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Charlie doesn't want his new house number to be divisible by 3 or 5. How many choices of house does he have?
Producing Zeros
problem

Producing zeros

Age
11 to 14
Challenge level
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Weekly Problem 10 - 2008
If the numbers 1 to 10 are all multiplied together, how many zeros are at the end of the answer?
Multiple years
problem

Multiple years

Age
11 to 14
Challenge level
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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?
Pairing Up
problem

Pairing up

Age
11 to 14
Challenge level
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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?
Stamp Collecting
problem

Stamp collecting

Age
11 to 14
Challenge level
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Last week, Tom and Sophie bought some stamps and they spent exactly £10. Can you work out how many stamps they bought?
Little Goldbach
problem

Little Goldbach

Age
11 to 14
Challenge level
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How many of the numbers 1 to 20 are not the sum of two primes?

Multiplication Table Puzzle
problem

Multiplication table puzzle

Age
11 to 14
Challenge level
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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?
Smallest Abundant Number
problem

Smallest abundant number

Age
11 to 14
Challenge level
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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?
Ones, Twos and Threes
problem

Ones, twos and threes

Age
11 to 14
Challenge level
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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?
AB Search
problem

AB search

Age
11 to 14
Challenge level
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The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?

Adjacent Factors
problem

Adjacent factors

Age
11 to 14
Challenge level
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Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?
Back of the Queue
problem

Back of the queue

Age
11 to 14
Challenge level
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Weekly Problem 48 - 2013
What is the remainder when the number 743589×301647 is divided by 5?
Digital Division
problem

Digital division

Age
11 to 14
Challenge level
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How many three digit numbers formed with three different digits from 0, 1, 2, 3 and 5 are divisible by 6?
One Short
problem

One short

Age
11 to 14
Challenge level
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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?
Red card blue card
problem

Red card blue card

Age
11 to 14
Challenge level
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Can you arrange the red and blue cards so that the rules are all followed?
Cakes and Buns
problem

Cakes and buns

Age
11 to 14
Challenge level
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Helen buys some cakes and some buns for her party. Can you work out how many of each she buys?
Find from Factors
problem

Find from factors

Age
11 to 14
Challenge level
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Weekly Problem 35 - 2006
A number has exactly eight factors, two of which are 21 and 35. What is the number?
Halloween Day
problem

Halloween day

Age
11 to 14
Challenge level
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One year there were exactly four Tuesdays and four Fridays in October. On what day of the week was Halloween.
Grandma's Cake
problem

Grandma's cake

Age
11 to 14
Challenge level
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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?
Jenny's Logic
problem

Jenny's logic

Age
11 to 14
Challenge level
filled star filled star empty star
Weekly Problem 52 - 2009
How did Jenny figure out that Tom's cards added to an even number?
17s and 23s
problem

17s and 23s

Age
11 to 14
Challenge level
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Can you form this 2010-digit number...
Common Remainder
problem

Common remainder

Age
11 to 14
Challenge level
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144 divided by n leaves a remainder of 11. 220 divided by n also leaves a remainder of 11. What is n?
HCF Expression
problem

HCF expression

Age
11 to 14
Challenge level
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Find out which two distinct primes less than $7$ will give the largest highest common factor of these two expressions.

Leap Monday
problem

Leap monday

Age
11 to 14
Challenge level
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Can you find the next time that the 29th of February will fall on a Monday?
Four or Five
problem

Four or five

Age
11 to 14
Challenge level
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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?
Seven from Nine
problem

Seven from nine

Age
11 to 14
Challenge level
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In how many ways can seven of the numbers 1-9 be chosen such that they add up to a multiple of 3?
Big Blackboard
problem

Big blackboard

Age
11 to 14
Challenge level
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Can you work out which numbers between 1 and 2016 have exactly two of 2, 3, 4 as factors?
Colossal Sum
problem

Colossal sum

Age
11 to 14
Challenge level
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What is the units digit in this sum of powers of 9?
Factor Sum
problem

Factor sum

Age
11 to 14
Challenge level
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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?
Missing Digit
problem

Missing digit

Age
11 to 14
Challenge level
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What digit must replace the star to make the number a multiple of 11?
Trailing Zeros
problem

Trailing zeros

Age
11 to 14
Challenge level
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How many zeros does 50! have at the end?
Cinema Costs
problem

Cinema costs

Age
11 to 14
Challenge level
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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 is cost this group of adults and children to see a film?
Powerful Finale
problem

Powerful finale

Age
11 to 14
Challenge level
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Weekly Problem 24 - 2017
What is the last digit of $3^{2011}$?
Indivisible
problem

Indivisible

Age
14 to 16
Challenge level
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Each time a class lines up in different sized groups, a different number of people are left over. How large can the class be?
Essential Supplies
problem

Essential supplies

Age
14 to 16
Challenge level
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Chocolate bars come in boxes of 5 or boxes of 12. How many boxes do you need to have exactly 2005 chocolate bars?
Punky Fish
problem

Punky fish

Age
14 to 16
Challenge level
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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?
End of a Prime
problem

End of a prime

Age
14 to 16
Challenge level
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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?
Threes and Fours
problem

Threes and fours

Age
14 to 16
Challenge level
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What is the smallest integer where every digit is a 3 or a 4 and it is divisible by both 3 and 4?
Peter's Primes
problem

Peter's primes

Age
14 to 16
Challenge level
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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?
Flora the Florist
problem

Flora the florist

Age
14 to 16
Challenge level
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Flora has roses in three colours. What is the greatest number of identical bunches she can make, using all the flowers?
Long List
problem

Long list

Age
14 to 16
Challenge level
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Weekly Problem 47 - 2017
How many numbers do I need in a list to have two squares, two primes and two cubes?
Sticky Fingers
problem

Sticky fingers

Age
14 to 16
Challenge level
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Ruth wants to puts stickers on the cuboid she has made from little cubes. Will she have any stickers left over?
Times and Square
problem

Times and square

Age
14 to 16
Challenge level
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Roger multiplies two consecutive integers and squares the result. Can you find the last two digits of his new number?
Eight Factors Only
problem

Eight factors only

Age
14 to 16
Challenge level
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We are given two factors of a number with eight factors. Can you work out the other factors of the number?
Divisible Palindrome
problem

Divisible palindrome

Age
14 to 16
Challenge level
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What is the sum of the digits of the largest 4-digit palindromic number which is divisible by 15?
Relative Time
problem

Relative time

Age
14 to 16
Challenge level
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Albert Einstein is experimenting with two unusual clocks. At what time do they next agree?
Powerful Zeros
problem

Powerful zeros

Age
14 to 16
Challenge level
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How many zeros are there at the end of $3^4 \times 4^5 \times 5^6$?
Coin Collection
problem

Coin collection

Age
14 to 16
Challenge level
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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?
Added Power
problem

Added power

Age
14 to 16
Challenge level
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How many integers $n$ are there for which $n$ and $n^3+3$ are both prime?
Square Sum
problem

Square sum

Age
14 to 16
Challenge level
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One of these numbers is the largest of nine consecutive positive integers whose sum is a perfect square. Which one is it?
Last-but-one
problem

Last-but-one

Age
14 to 16
Challenge level
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What is the last-but-one digit of 99! ?
Triangular Algebra
problem

Triangular algebra

Age
14 to 16
Challenge level
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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?
Three Primes
problem

Three primes

Age
14 to 16
Challenge level
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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?
Square LCM
problem

Square LCM

Age
14 to 16
Challenge level
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Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?

Factorised Factorial
problem

Factorised factorial

Age
14 to 16
Challenge level
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Weekly Problem 17 - 2010
The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?
Factor List
problem

Factor list

Age
14 to 16
Challenge level
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Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?
Adding a square to a cube
problem

Adding a square to a cube

Age
14 to 16
Challenge level
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If you take a number and add its square to its cube, how often will you get a perfect square?
Rational Integer
problem

Rational integer

Age
14 to 16
Challenge level
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Weekly Problem 39 - 2012
For how many values of $n$ are both $n$ and $\frac{n+3}{n−1}$ integers?
Square Product
problem

Square product

Age
14 to 16
Challenge level
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Weekly Problem 10 - 2011
Will this product give a perfect square?
Cancelling Fractions
problem

Cancelling fractions

Age
14 to 16
Challenge level
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Can you find a fraction with the following properties?
Fortunate Inflation
problem

Fortunate inflation

Age
14 to 16
Challenge level
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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?
Super Computer
problem

Super computer

Age
14 to 16
Challenge level
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What is the units digit of the given expression?
Primes and Six
problem

Primes and six

Age
14 to 16
Challenge level
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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 $36$.
Integer Indices
problem

Integer indices

Age
14 to 16
Challenge level
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From this sum of powers, can you find the sum of the indices?
Leftovers
problem

Leftovers

Age
14 to 16
Challenge level
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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 $n+4$?