Problems

Tommy has written 6 letters and addressed 6 envelopes. He then forgot which letter goes where and put them randomly such that no letter goes in the right envelope. In how many ways can he do this?

Annie and Hanna are preparing some Christmas baubles. They want to paint each bauble all in one colour. They have $7$ different colours of paint and $26$ baubles to paint. In how many ways can they do this? Two ways are considered the same if the numbers of baubles of each colour are the same. Each bauble has to be painted but not all the colours need to be used.

An $8\times 8$ square is divided into $1\times 1$ cells. It is covered with right-angled isosceles triangles (two triangles cover one cell). There are 64 black and 64 white triangles. We consider "regular" coverings - such that every two triangles having a common side are of a different colour. How many "regular" covers are there?

You are given a pentagon $ABCDE$ such that $AB=BC=CD=DE$, and $\mathrm{\angle }B=\mathrm{\angle }D={90}^{\text{circe}}$. Show how the plane can be tiled with pentagons equal to the given one.

Ms Jones vacuums her car every 2 days, she washes her car every 7 days and polishes it every 52 days. The last time she did all three types of cleaning on one day was on the 13th of March last year. What time will she do it again?

The numbers $a$ and $b$ are integers and $a>b$. Show that the gcd of $a$ and $b$ is equal to the gcd of $b$ and $a-b$.

A brave witch is out there hunting monsters for coin. She noticed that every 5th monster she encounters has wings, every 16th has a fiery breath, every 6th has fangs and every 14th has a pile of treasure. Now, the only monster with wings, fiery breath, fangs and a pile of treasure is a dragon and witches don’t hunt dragons. Assuming that the witch has just met a dragon, how many monsters will she have to hunt to meet another one?

Let $a=8\times 9^2\times {31}^2\times 7$ and $b=7^2\times 2^3\times 3^6\times {23}^2$. Find their greatest common divisor and least common multiple.

A number $n$ is an integer. Show that the gcd of $12n+9$ and $9n+6$ is $3$.

The gcd of numbers $a$ and $b$ is $72$. What can be their smallest possible product? What could be their greatest possible product?