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 \(12\).
By checking which of the following numbers are divisible by \(11\): \[121,\,143,\,286,\,235,\,473,\,798,\,693,\,576,\,748\] can you find and prove a condition that is enough to guarantee a 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?