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.
Let \(p\) be a prime number greater than \(3\). Prove that \(p^2-1\) is divisible by \(12\).
Split the numbers from \(1\) to \(9\) into three triplets such that the sum of the three numbers in each triplet is prime. For example, if you split them into \(124\), \(356\) and \(789\), then the triplet \(124\) is correct, since \(1+2+4=7\) is prime. But the other two triples are incorrect, since \(3+5+6=14\) and \(7+8+9=24\) are not prime.
Let \(p\), \(q\) and \(r\) be distinct primes at least \(5\). Can \(p^2+q^2+r^2\) be prime? If yes, then give an example. If no, then prove it.
Is there a divisibility rule for \(2^n\), where \(n = 1\), \(2\), \(3\), . . .? If so, then explain why the rule works.
Can you come up with a divisibility rule for \(5^n\), where \(n=1\), \(2\), \(3\), . . .? Prove that the rule works.