Let $n$ be the largest integer for which $14 n$ has exactly 100 digits. Counting from right to left, what is the 68 th digit of $n$ ?
(A) 1
(B) 2
(C) 4
(D) 5
(E) 8
The largest integer with exactly 100 digits is the integer that consists of 100 copies of the digit 9 . This integer is equal to $10^{100}-1$.

Therefore, we want to determine the largest integer $n$ for which $14 n \leq 10^{100}-1$.

This is the same as trying to determine the largest integer $n$ for which $14 n<10^{100}$, since $14 n$ is an integer.

We want to find the largest integer $n$ for which $n<\frac{10^{100}}{14}=\frac{10}{14} \times 10^{99}=\frac{5}{7} \times 10^{99}$.

This is equivalent to calculating the number $\frac{5}{7} \times 10^{99}$ and rounding down to the nearest integer. Put another way, this is the same as calculating $\frac{5}{7} \times 10^{99}$ and truncating the number at the decimal point.

The decimal expansion of $\frac{5}{7}$ is $0 . \overline{714285}$. (We can see this either using a calculator or by doing long division.)

Therefore, the integer that we are looking for is the integer obtained by multiplying $0 . \overline{714285}$ by $10^{99}$ and truncating at the decimal point.

In other words, we are looking for the integer obtained by shifting the decimal point in $0 . \overline{714285}$ by 99 places to the right, and then ignoring everything after the new decimal point.

Since the digits in the decimal expansion repeat with period 6 , then the integer consists of 16 copies of the digits 714285 followed by 714 . (This has $16 \times 6+3=99$ digits.)

In other words, the integer looks like $714285714285 \cdots 714285714$.

We must determine the digit that is the 68th digit from the right.

If we start listing groups from the right, we first have 714 (3 digits) followed by 11 copies of 714285 ( 66 more digits). This is 69 digits in total.

Therefore, the " 7 " that we have arrived at is the 69th digit from the right.

Moving one digit back towards the right tells us that the 68th digit from the right is 1.

ANSWER: (A)