Math: Find the digit

Find the Digit

Let N be the largest integer such that N and 7N both have 100 digits.

What is the 50th digit of n?

Solution

If N is the largest number such that N and 7N both have 100 digits, then 7N must be the largest multiple of 7 that has 100 digits. We can therefore find N just by dividing 7 into 9999...99 (100 9's).

By simple long division, we find that N starts off with repeated blocks of 142857. The first, seventh, thirteenth, ..., forty-ninth digits must therefore be 1's, and so the fiftieth digit is a 4.