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 \(1 + 2 + \dots + 2018\)?
What is a remainder in division by \(3\) of the number \(8^{2019}\)?
Show that a number \(3333333333332\) is not a perfect square (without using a calculator).
What is a remainder in division by \(3\) of the number \(5^{21} + 17^6 \times 7^{2019}\)?
Show that the sum of any three consecutive integers is divisible by \(3\).
In a country far far away, there are only two types of coins: 1 crown and 3 crowns coins. Molly had a bag with only 3 crown coins in it. She used some of these coins to buy herself hat and she got one 1 crown coin back. The next day, all of her friends were jealous of her hat, so she decided to buy identical hats for them. She again only had 3 crown coins in her purse, and she used them to pay for 7 hats. Show that she got a single 1 crown coin back.