Problems

Let $n$ be a nonnegative integer. What is the gcd of $12n+9$ and $9n+6$?

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

a) Two numbers, $a$ and $b$, are relatively prime and their product is equal to $60$. What could these numbers be? Find all the possibilities.
b) The GCD of two numbers, $c$ and $d$, is $18$ and their product is $2^4\times 3^5\times 7$. What could these numbers be? Find all the answers.

a) Can you measure $5$ litres of milk using two buckets of volumes $4$ and $11$ litres respectively?
b) Can you measure $7$ litres of milk using buckets of volumes $8$ and $12$ litres respectively?

a) A mighty dragon has several rubies in his treasure. He is able to divide the rubies into groups of $3$, $5$ or $11$. How many rubies does he have, if we know that is fewer than $200$?
b) The same dragon also has some emeralds. He is $6$ emeralds short to be able to divide them into groups of $13$, one emerald short to be able to divide them into groups of $5$, but if he wants to divide them into groups of $8$, he is left with one emerald. How many emeralds does he have if we know it is fewer than $500$?

Let $n$ be a natural number. Show that the fraction $\frac{21n+4}{14n+3}$ is irreducible, i.e. it cannot be simplified.

Let $m$ and $n$ be two positive integers with $m<n$ such that $$gcd(m,n)+\text{lcm}(m,n)=m+n.$$ Show that $m$ divides $n$.

The numbers $x,a,b$ are natural. Show that $gcd(x^a-1,x^b-1)=x^{gcd(a,b)}-1$.

Let $p$ be a prime number bigger than $3$. Prove that $p^2-1$ is a multiple of 24.

Is it possible to draw $K_5$ without intersecting edges on a Möbius band? Recall that $K_5$ is the complete graph on $5$ vertices. That is, $5$ points with an edge between every pair of different points.