2
$\begingroup$

Number theory question. Not sure how to approach this one, does the ... mean 0s or any number?

  • 4
    I assume it means $0's$. Under that assumption, Hint: such a number is of the form $10^n+14$ for some $n$. – 2017-02-08
  • 3
    I assume the dots mean zeros. Then if $10\ldots014$ were divisible by $7$, then $10\ldots 000$ would also be divisible by $7$. Does that seem right? (Maybe think prime factorization?) – 2017-02-08
  • 3
    Just to say: $105014$ is divisible by $7$, so the dots can't represent "any number". – 2017-02-08
  • 0
    $$10^k+14\equiv 0\pmod{7}\quad\Longrightarrow\quad 3^k\equiv 0\pmod{7}$$ – 2017-02-08

2 Answers 2

5

This number is of the form $10^n + 14$ (for $n \ge 4$). Of course 7 divides 14, but $7$ does not divide $10^n$, whose prime factorization is $2^n 5^n$.

1

Suppose $100 \ldots 014$ is divisible by $7$, then so is $100\ldots 014 - (2 \times 7) = 100\ldots 014 - 14 = 100 \ldots 000$. However, the only prime factors of $100 \ldots 000$ are $2,5$ (it is $10^1$ for some $n > 0$), hence $7$ does not divide this number, giving a contradiction.

I do not see any other meaning of the $\ldots$ in the question. The $...$ cannot mean something arbitrary : for example $105014$ as given above.