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?
Can a \(300\)-digit number, with one hundred \(0\)s, one hundred \(1\)s, and one hundred \(2\)s among its digits, be a square number?
Robinson Crusoe’s friend Friday was looking at \(3\)-digit numbers with the same first and third digits. He soon noticed that such number is divisible by \(7\) if the sum of the second and the third digits is divisible by \(7\). Prove that he was right.
Prove the following divisibility rule by 37: divide the number starting from the right end of the number into blocks of three digits. Now, the original number is divisible by 37 if the sum of all three digit numbers obtained in this way is divisible by 37.
(It might be the case that the number of digits is not divisible by 3, and you cannot divide the original number into blocks of three digits. To overcome this problem we allow a block of three digits to start from 0, for example number 2345678 should be divided into blocks of three digits as 002, 345, and 678.)
2016 digits are written in a circle. It is known, that if you make a number reading the digits clockwise, starting from some particular place, then the resulting 2016-digit number is divisible by 27. Show that if you start from some other place, and moving clockwise make up another 2016-digit number, then this new number is also divisible by 27.
We call a \(10\)-digit number interesting if it is divisible by \(11111\), and all its digits are different. How many interesting numbers does there exist?
Note that a number \(k = a_0 + 10a_1 + \dots +10^9 a_9\) is divisible by \(11111\) if and only if a number \(m = (a_0+a_5) +10(a_1+a_6) + \dots + 10^4 (a_4+a_9)\) is also divisible by \(11111\). This is because \(100000=1+9 \times 11111\) and we subtract \(99999 (a_5 + 10a_6 + 100a_7 + 1000a_8 +10000a_9)\) from the original number.
Louise is confident that all her classmates have different number of friends. Is she right?
There are 100 cities all connected by roads. Each city has 6 roads coming in (or going out). How many roads do connect those cities?
There are 15 cities in a country named The Country of Fifteen Cities. The king ordered his main architect to build roads in such a way that each city was connected with other cities by exactly 5 roads, otherwise he would hang the architect. Do you think that the architect can accomplish the task or should he flee that country immediately?
The architect decided to flee The Country of 15 Cities and began to travel around the world. He arrived to a country, where every city had exactly 3 roads going to and from it. Can there be all together 100 roads in that country?