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 \(\angle B = \angle D = 90^\circe\). 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 coins. 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. Suppose that the witch has just met a dragon and that every dragon has wings, firey breath, fangs and a pile of treasure. 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.
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?