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.
Show that numbers \(12n+1\) and \(12n+7\) are relatively prime.
If natural numbers \(a,b\) and \(c\) are lengths of the sides of a right triangle (such that \(a^2+b^2=c^2\)), show that at least one of these numbers is divisible by \(3\).