The numbers \(2^{2018}\) and \(5^{2018}\) are expanded and their digits are written out consecutively on one page. How many digits are on the page?
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?
In the context of Example 14.2 what is the answer if we have five numbers instead of four? (i.e., can we get four distinct prime numbers then?)
The number \(b^2\) is divisible by \(8\). Show that it must be divisible by \(16\).
Find a number which:
a) It is divisible by \(4\) and by \(6\), is has a total of 3 prime factors, which may be repeated.
b) It is divisible by \(6, 9\) and \(4\), but not divisible by \(27\). It has \(4\) prime factors in total, which may be repeated.
c) It is divisible by \(5\) and has exactly \(3\) positive divisors.