Solve the equation with natural numbers \(1 + x + x^2 + x^3 = 2y\).
Prove there are no natural numbers \(a\) and \(b\), such as \(a^2 - 3b^2 = 8\).
Prove there are no integer solutions for the equation \(x^2 + 1990 = y^2\).
Let \(p\) and \(q\) be two prime numbers such that \(q = p + 2\). Prove that \(p^q + q^p\) is divisible by \(p + q\).
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?