Problems

Let us introduce the notation – we denote the product of all natural numbers from 1 to $n$ by $n!$. For example, $5!=1\times 2\times 3\times 4\times 5=120$.

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?

For what natural numbers $a$ and $b$ is the number ${\log }_{a}b$ rational?

In a row there are 20 different natural numbers. The product of every two of them standing next to one another is the square of a natural number. The first number is 42. Prove that at least one of the numbers is greater than 16,000.