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?
At the end of the term, Billy wrote out his current singing marks in a row and put a multiplication sign between some of them. The product of the resulting numbers turned out to be equal to 2007. What is Billy’s term mark for singing? (The marks that he can get are between 2 and 5, where 5 is the highest mark).
The product of two natural numbers, each of which is not divisible by 10, is equal to 1000. Find the sum of these two numbers.
Write in terms of prime factors the numbers 111, 1111, 11111, 111111, 1111111.
Try to get one billion \(1000000000\) by multiplying two whole numbers, in each of which there cannot be a single zero.