# Divisibility of consecutive natural numbers

I found this task followed by a hint,
that I should try to apply Chinese remainder theorem to that:

Prove, that there exist 2012 consecutive natural numbers,
which satisfy that every one of them is divisible by a cube of a natural number $\ge$ 2.

The problem is, I don't really see how to use the theorem above. Can anyone help?

• Every natural number is divisible by $1^3$. – Chris Eagle Aug 25 '12 at 20:35
• i think that as @ Chris Eagle said only solution is $1$,otherwise you have to know that,among these $2012$ consecutive numbers,some of them is prime,which can't be divides by any number or even by it's cube – dato datuashvili Aug 25 '12 at 20:38
• @dato That's not what Chris said ("only" solution). Chris is getting at the $\ge1$ thing (which should be written $>1$ or else the problem is trivial). What makes you think a stretch of $2012$ consecutive numbers necessarily contains a prime? – anon Aug 25 '12 at 20:39
• @anon i understood what he said,just i have added about prime numbers – dato datuashvili Aug 25 '12 at 20:40
• consecutive number i think means any number +1 right? – dato datuashvili Aug 25 '12 at 20:42

## 1 Answer

Look at the system of congruences $x\equiv 0 \bmod{2^3}$, $\,x+1\equiv 0\bmod{3^3}$, $\,x+2\equiv 0\bmod{5^3}$, $x+3\equiv 0\bmod{7^3}$, $\,x+4\equiv 0\bmod{11^3}$, and so on.

• How does that work with $x\equiv 0\bmod{3^3}$ and $x+3\equiv 0\bmod{9^3}$? – SiliconCelery Aug 25 '12 at 21:00
• @SiliconCelery Where are you getting $9^3$ from? The moduli that Andre is using are the cubes of prime numbers, and $9$ is not prime. It would be $x+3\equiv0~\bmod 7^3$, since $7$ is the next prime number after $5$. – anon Aug 25 '12 at 21:07
• @SiliconCelery: Thanks, you pointed out a possible ambiguity in what I meant by and so on. It is the primes cubed. I changed things so that this is clearer. – André Nicolas Aug 25 '12 at 21:08
• I'd add that the fact that the numbers on the right hand side are cubes of distinct primes and therefore all co-prime means the Chinese remainder theorem will find a solution. Finding the smallest solution seems very difficult. – gnasher729 May 9 '14 at 15:31
• @gnasher729: You are right, the above construction based on CRT will undoubtedly produce a number $x$ enormously larger than the smallest possible number with the required property. Getting even basic size information about the smallest number seems pretty hopeless. – André Nicolas May 9 '14 at 15:39