Factors, Multiple and Primes - Short Problems

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?

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

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

Weekly Problem 49 - 2013
What is the value of 2000 + 1999 × 2000?

Weekly Problem 24 - 2006
How many of the rearrangements of the digits 1, 3 and 5 give prime numbers?

Weekly Problem 26 - 2014
Which of the given numbers is divisible by 6?

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

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

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

Weekly Problem 10 - 2008
If the numbers 1 to 10 are all multiplied together, how many zeros are at the end of the answer?

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 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?

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

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

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?

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

Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?

Weekly Problem 48 - 2013
What is the remainder when the number 743589×301647 is divided by 5?

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

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?

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

Weekly Problem 35 - 2006
A number has exactly eight factors, two of which are 21 and 35. What is the number?

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

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

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?

Weekly Problem 52 - 2009
How did Jenny figure out that Tom's cards added to an even number?

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?

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?

Can you form this 2010-digit number...

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

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

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?

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

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

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

What is the units digit in this sum of powers of 9?

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?

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

How many zeros does 50! have at the end?

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?

Weekly Problem 24 - 2017
What is the last digit of $3^{2011}$?

Each time a class lines up in different sized groups, a different number of people are left over. How large can the class be?

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

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?

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?

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

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

Weekly Problem 47 - 2017
How many numbers do I need in a list to have two squares, two primes and two cubes?

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

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

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

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

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

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

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?

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

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

What is the last-but-one digit of 99! ?

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?

Weekly Problem 17 - 2010
The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?

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

Weekly Problem 39 - 2012
For how many values of $n$ are both $n$ and $\frac{n+3}{n−1}$ integers?

Weekly Problem 10 - 2011
Will this product give a perfect square?

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

What is the units digit of the given expression?

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 $36$.

Can you find a fraction with the following properties?

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?

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

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 $n+4$?

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?

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