Draw how Robinson Crusoe should use pegs, ropes, and sliding rings to tie his goat in order for the goat to graze grass in the shape of a semicircle.
Draw a picture how Robinson could have used pegs and ropes to tie the wolf and the goat so that the goat grazed an area in the shape of a ring (like a disc with a hole in the middle).
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 \(3\): a number is divisible by \(3\) if and only if the sum of its digits is divisible by \(3\).
Sophia is playing the following game: she chooses a whole number, and then she writes down the product of all the numbers from \(1\) up to the number she chose. For example, if she chooses \(5\), then she writes down \(1\times 2 \times 3 \times 4 \times 5\). What is the smallest number she can choose for her game, such that the result she gets in the end is divisible by \(2024\)?
While studying numbers and their properties, Robinson came across a three-digit prime number whose last digit equals the sum of the first two digits. What are the options for the last digit of this number, given that none of its digits is zero?
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 and his friend Friday learned about divisibility rules, Friday decided to proposed his own rule:
if a number is divisible by \(27\), then the sum of its digits is also divisible by \(27\).
Was Friday 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?