A composite number containing exactly 1993 digits "1" and one digit "7" [closed]

elementary-number-theory

You are correct for the first.

Hint: For the second, if the $7$ is in the $10^n$ place your number is $1111...1 (1994$ ones) $+ 6\cdot 10^n$. What is the remainder of dividing the number with all the $1$s by $7$? Can you find an $n$ that makes the sum divisible by $7?$

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