Two numbers are \(a = 2 \times 3^5 \times 31^2 \times 7\) and \(b= 7^2 \times 2^4 \times 3^2 \times 29^2\). Find their greatest common divisor and least common multiple.
A valiant adventurer enters a dragon’s cave looking for the Holy Grail. She knows that Holy Grail is a chalice that is tall, made of gold, has encrusted rubies, and has an ancient inscription written on it. Upon entering, the knight discovers a long corridor of chalices, all marked with natural numbers starting from 1. He examines the first chalices and discovers, that every 10th chalice is tall, every 15th is made of gold, every 28th has encrusted rubies and every 27th has an ancient description. Assuming that is universally true for all the chalices in the cave, which chalice should the adventurer check so she doesn’t waste too much time checking all of them?
The gcd of numbers \(a\) and \(b\) is \(40\). What can be their smallest possible product? What could be their greatest possible product?
a) While visiting Cape Verde, Pirate Jim and Pirate Bob bought several chocolate chip cookies each. Jim paid 93 copper coins for his cookies and Bob paid 102 copper coins. What could be the price of a single cookie if it is a natural number?
b) Captain Hook and Captain Kid bought several tricorn hats each. Captain Hook paid 6 silver coins more than Captain Kid. What could be the price of a tricorn hat if it is an integer?
Show that \(a-b\) is divisible by the greatest common divisor of \(a\) and \(b\).
a) Can you measure \(6\) litres of water using two buckets of volumes \(7\) and \(10\) litres respectively?
b) Can you measure \(7\) litres of water using buckets of volumes \(9\) and \(12\) litres respectively?
a) King Haggard has a velvet pouch filled with diamonds. He can divide these diamonds into 3 equal piles, 4 equal piles, or 5 equal piles. How many diamonds does he have if it is known that his collection contains less than 100 diamonds in total?
b) King Haggard has a stash of gold coins. He is one coin short of being able to divide these coins into 4 equal piles, or 5 equal piles, or 6 equal piles, or 7 equal piles. How many coins does he have if he has fewer than 500?
Suppose that \(p\) is a prime number.
a) How many numbers that are less than \(p\) are relatively prime to it?
b) How many numbers that are less than \(p^2\) are relatively prime to it?
I have written \(5\) non-prime numbers on a piece of paper and hidden it in a safe locker. Every pair of these numbers is relatively prime. Show that at least one of these numbers has to be larger than a \(100\).
What is a remainder in division by \(3\) of the sum \(1 + 2 + \dots + 2018\)?