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# Factors, Multiple and Primes - Short Problems

### Producing Zeros

### Factor Trio

### Reversible Primes

### Multiple Years

### Product 100

### Pairing Up

### Tricky Customer

### Calculation 2000

### Stamp Collecting

### Prime Order

### Almost a Million

### Divisible Digits

### Little Goldbach

### What's on the Back?

### Multiplication Table Puzzle

### One Short

### Grandma's Cake

### Cakes and Buns

### Find from Factors

### Jenny's Logic

### Back of the Queue

### Digital Division

### Smallest Abundant Number

### AB Search

### Adjacent Factors

### Ones, Twos and Threes

### Halloween Day

### Red Card Blue Card

### Cinema Costs

### Seven from Nine

### HCF Expression

### Powerful Finale

### Factor Sum

### Leap Monday

### Missing Digit

### Four or Five

### Big Blackboard

### Common Remainder

### Peter's Primes

### Indivisible

### Flora the Florist

### Punky Fish

### End of a Prime

### Threes and Fours

### Essential Supplies

### Powerful Zeros

### Long List

### Divisible Palindrome

### Triangular Algebra

### Relative Time

### Times and Square

### Coin Collection

### Added Power

### Sticky Fingers

### Eight Factors Only

### Square Sum

### Primes and Six

### Adding a Square to a Cube

### Cancelling Fractions

### Factor List

### Three Primes

### Fortunate Inflation

### Leftovers

### Square LCM

### Rational Integer

### Factorised Factorial

### Square Product

### Integer Indices

## You may also like

### Seven Squares - Group-worthy Task

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Links to the NRICH website Home page

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30 April (Primary), 1 May (Secondary)

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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.

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Which of the numbers shown is the product of exactly 3 distinct prime factors?

Age 11 to 14

ShortChallenge Level

How many two-digit primes are there between 10 and 99 which are also prime when reversed?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

The product of four different positive integers is 100. What is the sum of these four integers?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Charlie doesn't want his new house number to be divisible by 3 or 5. How many choices of house does he have?

Age 11 to 14

ShortChallenge Level

Weekly Problem 49 - 2013

What is the value of 2000 + 1999 Ã— 2000?

Age 11 to 14

ShortChallenge Level

Last week, Tom and Sophie bought some stamps and they spent exactly Â£10. Can you work out how many stamps they bought?

Age 11 to 14

ShortChallenge Level

Weekly Problem 24 - 2006

How many of the rearrangements of the digits 1, 3 and 5 give prime numbers?

Age 11 to 14

ShortChallenge Level

Weekly Problem 26 - 2014

Which of the given numbers is divisible by 6?

Age 11 to 14

ShortChallenge Level

Can you find the missing digits, given that the number is divisible by 3, 4, 5 and 6?

Age 11 to 14

ShortChallenge Level

How many of the numbers 1 to 20 are not the sum of two primes?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Helen buys some cakes and some buns for her party. Can you work out how many of each she buys?

Age 11 to 14

ShortChallenge Level

Weekly Problem 35 - 2006

A number has exactly eight factors, two of which are 21 and 35. What is the number?

Age 11 to 14

ShortChallenge Level

Weekly Problem 52 - 2009

How did Jenny figure out that Tom's cards added to an even number?

Age 11 to 14

ShortChallenge Level

Weekly Problem 48 - 2013

What is the remainder when the number 743589Ã—301647 is divided by 5?

Age 11 to 14

ShortChallenge Level

How many three digit numbers formed with three different digits from 0, 1, 2, 3 and 5 are divisible by 6?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

One year there were exactly four Tuesdays and four Fridays in October. On what day of the week was Halloween.

Age 11 to 14

ShortChallenge Level

Can you arrange the red and blue cards so that the rules are all followed?

Age 11 to 14

ShortChallenge Level

Weekly Problem 41 - 2009

At a cinema a child's ticket costs Â£4.20 and an adult's ticket costs Â£7.70. How much did is cost this group of adults and children to see a film?

Age 11 to 14

ShortChallenge Level

In how many ways can seven of the numbers 1-9 be chosen such that they add up to a multiple of 3?

Age 11 to 14

ShortChallenge Level

Find out which two distinct primes less than $7$ will give the largest highest common factor of these two expressions.

Age 11 to 14

ShortChallenge Level

Weekly Problem 24 - 2017

What is the last digit of $3^{2011}$?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Can you find the next time that the 29th of February will fall on a Monday?

Age 11 to 14

ShortChallenge Level

What digit must replace the star to make the number a multiple of 11?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Can you work out which numbers between 1 and 2016 have exactly two of 2, 3, 4 as factors?

Age 11 to 14

ShortChallenge Level

144 divided by n leaves a remainder of 11. 220 divided by n also leaves a remainder of 11. What is n?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

Flora has roses in three colours. What is the greatest number of identical bunches she can make, using all the flowers?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

What is the smallest integer where every digit is a 3 or a 4 and it is divisible by both 3 and 4?

Age 14 to 16

ShortChallenge Level

Chocolate bars come in boxes of 5 or boxes of 12. How many boxes do you need to have exactly 2005 chocolate bars?

Age 14 to 16

ShortChallenge Level

How many zeros are there at the end of $3^4 \times 4^5 \times 5^6$?

Age 14 to 16

ShortChallenge Level

Weekly Problem 47 - 2017

How many numbers do I need in a list to have two squares, two primes and two cubes?

Age 14 to 16

ShortChallenge Level

What is the sum of the digits of the largest 4-digit palindromic number which is divisible by 15?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

Albert Einstein is experimenting with two unusual clocks. At what time do they next agree?

Age 14 to 16

ShortChallenge Level

Roger multiplies two consecutive integers and squares the result. Can you find the last two digits of his new number?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

How many integers $n$ are there for which $n$ and $n^3+3$ are both prime?

Age 14 to 16

ShortChallenge Level

Ruth wants to puts stickers on the cuboid she has made from little cubes. Will she have any stickers left over?

Age 14 to 16

ShortChallenge Level

We are given two factors of a number with eight factors. Can you work out the other factors of the number?

Age 14 to 16

ShortChallenge Level

One of these numbers is the largest of nine consecutive positive integers whose sum is a perfect square. Which one is it?

Age 14 to 16

ShortChallenge 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 $36$.

Age 14 to 16

ShortChallenge Level

If you take a number and add its square to its cube, how often will you get a perfect square?

Age 14 to 16

ShortChallenge Level

Can you find a fraction with the following properties?

Age 14 to 16

ShortChallenge Level

Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge 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 $n+4$?

Age 14 to 16

ShortChallenge Level

Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?

Age 14 to 16

ShortChallenge Level

Weekly Problem 39 - 2012

For how many values of $n$ are both $n$ and $\frac{n+3}{nâˆ’1}$ integers?

Age 14 to 16

ShortChallenge Level

Weekly Problem 17 - 2010

The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?

Age 14 to 16

ShortChallenge Level

Weekly Problem 10 - 2011

Will this product give a perfect square?

Age 14 to 16

ShortChallenge Level

From this sum of powers, can you find the sum of the indices?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?