Can one cut a square into (a) one 30-gon and five pentagons? (b) one 33-gon and three 10-gons?
A young mathematician had quite an odd dream last night. In his dream he was a knight on a \(4\times4\) board. Moreover, he was moving like a knight moves on the usual chessboard. In the morning he could not remember what was actually happening in his dream, though the young mathematician is pretty sure that either
(a) he has passed exactly once through all the cells of the board except for the one at the bottom leftmost corner, or
(b) he has passed exactly once through all the cells of the board.
For each possibility examine if it could happen or not.
Is it true that if a natural number is divisible by 4 and by 6, then it must be divisible by \(4\times6=24\)?
And what if a natural number is divisible by 5 and by 7? Should it be divisible by 35?
The number \(A\) is not divisible by 3. Is it possible that the number \(2A\) is divisible by 3?
List the first \(10\) prime numbers and write the prime decomposition of \(2910\).
Lisa knows that \(A\) is an even number. But she is not sure if \(3A\) is divisible by 6. What do you think?
George divided number \(a\) by number \(b\) with the remainder \(d\) and the quotient \(c\). How will the remainder and the quotient change if the dividend and the divisor are increased by a factor of 3?
Let us introduce the notation – we denote the product of all natural numbers from 1 to \(n\) by \(n!\). For example, \(5!=1\times2\times3\times4\times5=120\).
a) Prove that the product of any three consecutive natural numbers is divisible by \(3!=6\).
b) What about the product of any four consecutive natural numbers? Is it always divisible by 4!=24?
Can a sum of three different natural numbers be divisible by each of those numbers?