Is it possible for \(n!\) to be written as \(2015000\dots 000\), where the number of 0’s at the end can be arbitrary?
Look again at the divisibility rule for \(3\). Can you come up with a divisibility rule for \(9\)?
A battle of the captains was held at a maths battle. The task was to write the smallest number such that it is divisible by 45 and consists of only 1’s and 0’s as digits. What do you think was the correct answer?
It is known that a natural number is three times bigger than the sum of its digits. Is it divisible by 27?
A number was left written on the white board after a maths class. The number consisted of one hundred 0’s, one hundred 1’s, and one hundred 2’s as digits. A cleaner was about to wipe it off when suddenly he saw a small comment written in a corner. The comment stated that the number was a square number. He fetched a sigh and wrote “it not a square number”. Why was he right?
A stoneboard was found on the territory of the ancient Greek Academia as a result of archaeological excavations.
The archeologists decided that this stoneboard belonged to a mathematician who lived in the 7th century BC. The list of unsolved problems was written on the stoneboard. The archaeologists became thrilled to solve the problems but got stuck on the fifth. They were looking for a 10-digit number. The number should consist of only different digits. Moreover, if you cross any 6 digits, the remaining number should be composite. Can you help the archeologists to figure out the answer?
Divide 15 walnuts into four groups, each group consisting of a different whole number of nuts.
In the following example with fractions replace “stars” with different natural numbers in order to obtain an identity: \[\frac{1}{*}+\frac{1}{*}=\frac{1}{*}+\frac{1}{*}.\]
Looking back at Example 12.1 what if we additionally require all differences to be less than the smallest of the three numbers?
(a) Divide 55 walnuts into four groups consisting of different number of nuts.
(b) Divide 999 walnuts into four groups consisting of different number of nuts.