A triple of natural numbers \(a,b,c\) such that \(a^2 + b^2 = c^2\) is called a Pythagorean triple. There are some small Pythagorean triples that are well-known, like \(3,4,5\) and \(5,12,13\). Let us have a look at the latter one. We can notice an interesting thing: not only \(5^2+12^2=13^2\), but also \(5^2 = 25 = 12+13\) and \(13-12=1\). Use that as an inspiration to find an idea of how to generate some more Pythagorean triples. Check if they are correct by plugging them into the equation \(a^2 +b^2 = c^2\).
Two semicircles and one circle were drawn on the sides of a right triangle. The circle whose centre is in the midpoint of the hypothenuse actually goes through the right angle corner – this is a general fact, but you don’t need to prove it here. If the two shorter sides of the triangle are \(3\) and \(4\), what is the total area of the red region?
The lengths of three sides of a right triangle are all integer numbers.
a) Show that one of them is divisible by \(3\).
b*) Show that one of them is divisible by \(5\).
A segment \(AB\) is a base of an isosceles triangle \(ABC\). A line perpendicular to the segment \(AC\) was drawn through point \(A\) – this line crosses an extension of the segment \(BC\) at point \(D\). There is also a point \(E\) somewhere, such that angles \(\angle ECB\) and \(\angle EBA\) are both right. Point \(F\) is on the extension of the segment \(AB\), such that \(B\) is between \(A\) and \(F\). We also know that \(BF = AD\). Show that \(ED =EF\).
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.