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.
Let \(p\) be a prime number greater than \(3\). Prove that \(p^2-1\) is divisible by \(12\).
Which of the following numbers are divisible by \(11\) and which are not? \[121,\, 143,\, 286, 235, \, 473,\, 798, \, 693,\, 576, \,748\] Can you write down and prove a divisibility rule which helps to determine if a three digit number is divisible by \(11\)?
You meet an alien, who you learn is thinking of a positive integer \(n\). They ask the following three questions.
“Am I the kind who could ask whether \(n\) is divisible by no primes other than \(2\) or \(3\)?"
“Am I the kind who could ask whether the sum of the divisors of \(n\) (including \(1\) and \(n\) themselves) is at least twice \(n\)?"
“Is \(n\) divisible by 3?"
Is this alien a Crick or a Goop?