[ Divisibility of numbers. General properties ] [ Examples and counterexamples. Constructions ]

A five-digit number is called indivisible if it cannot be factored into the product of two three-digit numbers.

What is the largest number of consecutive indivisible five-digit numbers?
Note that all numbers divisible by 100 are decomposable. Therefore, there cannot be more than 99 consecutive non-decomposable numbers. On the other hand, between the numbers $100 \cdot 100$ and $100 \cdot 101$, all numbers are decomposable, and there are exactly 99 of them.

## Answer

99. 

All coefficients of the polynomial $P(x)$ are integers. It is known that $P(1)=1$ and that $P(n)=0$ for some natural number $n$. Find $n$.

## Solution

By the theorem of Bezout for integer polynomials (see the solution of problem $\underline{35562})-1=P(n)-P(1)$ is divisible by $n-1$. Therefore, $n-1= \pm 1$, which means $n=2$.

## Answer

$n=2$.