Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a semicircle
In his twelfth year on the island Robinson Crusoe managed to tame a wolf, and used him as a guard dog for his goat. He used to tie the wolf with ropes to pegs and other ropes in such a way that there was always a safe place for the goat to escape. Subsequently, he studied the shapes the goat was grazing on the ground.
Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of a ring
Think of other shapes Robinson’s goat can graze without a wolf, or with a wolf tied nearby. What if Robinson managed to tame several wolves and used them as guard dogs? Can two tied wolves keep an untied goat in a triangle? Can you think of other shapes you can create with Robinson’s goat and wolves?
Prove the divisibility rule for \(3\): the number is divisible by \(3\) if and only if the sum of its digits is divisible by \(3\).
Find the smallest \(k\) such that \(k!\) (\(k!= k\times(k-1)\times \ldots \times 1\)) is divisible by \(2024\).
While studying numbers and its properites, Robinson came across a 3-digit prime number with the last digit being equal to the sum of the first two digits. What was the last digit of that number if among the number did not have any zeros among it’s digits?
Prove the divisibility rule for \(4\): a number is divisible by \(4\) if and only if the number made by the last two digits of the original number is divisible by \(4\);
Can you come up with a divisibility rule for \(8\)?
Robinson found a chest with books and instruments after the ship wreck. Not all the books were in readable condition, but some of the books he managed to read. One sentence read “72 chickens cost *619* p”. (The starred digits were not readable). He has not tasted a chicken for quite some time, and it was pleasant to imagine a properly cooked chicken in front of him. He also was able to decipher the cost of one chicken. Can you?
When Robinson Crusoe’s friend and assistant named Friday learned about divisibility rules, he was so impressed that he proposed his own rule:
a number is divisible by 27 if the sum of it’s digits is divisible by 27.
Was he right?
One day Friday multiplied all the numbers from 1 to 100. The product appeared to be a pretty large number, and he added all the digits of that number to receive a new smaller number. Even then he did not think the number was small enough, and added all the digits again to receive a new number. He continued this process of adding all the digits of the newly obtained number again and again, until finally he received a one-digit number. Can you tell what number was it?