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?
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 \[\gcd(m,n)+\lcm(m,n)=m+n.\] 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.