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.
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.
George knows a representation of number “8” as the sum of its divisors in which only divisor “1” appears twice: \[8=4+2+1+1.\] His brother showed George that such representation exists for number “16” as well: \[16=8+4+2+1+1.\] He apologies for forgetting an example considering number “32” but he is sure once he saw such representation for this number.
(a) Help George to work out a suitable representation for number “32”;
(b) Can you think of a number which has such representation consisting of 7 terms?
(c) Of 11 terms?
(d) Can you find a number which can be represented as a sum of its divisors which are all different (pay attention that we don’t allow repeating digit “1” twice!)?
(e) What if we require this representation to consist of 11 terms?
Jennifer draws a hexagon, and a line passing through two of its vertices. It turns out one of the figures in which the original hexagon is divided is a heptagon. Show an example of a hexagon and a line for which it is true.
There are four numbers written in a row. The first number is \(100\). It is known that if we divide the first number by the second number we will get a prime number as a result, if we the second number by the third number we will get a prime number, and if we divide the third number by the fourth number we will also get a prime number. Can all the resulting prime numbers be distinct?
George claims that he knows two numbers such that their quotient is equal to their product. Can we believe him? Prove him wrong or provide a suitable example.