25. Let $n$ be the smallest integer for which $7 n$ has 2016 digits.
What is the units digit of $n$ ?
A 0
B 1
C 4
D 6
E 8
SolUtion
D
The smallest positive integer which has 2016 digits is the number $10^{2015}$, which is written with the digit 1 followed by 2015 copies of the digit 0 . It follows that we require $n$ to be the least positive integer for which $7 n \geq 10^{2015}$. Therefore $n$ is the least positive integer such that
$$
n \geq \frac{10^{2015}}{7} \text {. }
$$

The decimal expansion of $\frac{1}{7}$ is the recurring decimal 0.142857 , with the block 142857 of six digits repeated for ever after the decimal point.
We have
$$
\frac{10^{2015}}{7}=10^{2015} \times \frac{1}{7} \text {. }
$$

The effect of multiplying by $10^{2015}$ is to move the decimal point 2015 places to the right. It follows that the decimal representation of $\frac{10^{2015}}{7}$ consists of the digits 142857 in repeated blocks, with 2015 of these digits coming before the decimal point.
Now $2015=6 \times 335+5$. Therefore the decimal expansion of $\frac{10^{2015}}{7}$ consists of 335 repetitions of the block 142857 followed by 14285 before the decimal point, with the digit 7 followed by an unending repetition of the block of digits 142857 after the decimal point.

That is,
$$
\frac{10^{2015}}{7}=142857142857 \ldots 14285714285.7142857
$$

It follows that the least integer $n$ such that $n \geq \frac{10^{2015}}{7}$ is
$$
142857142857 \ldots 14285714286,
$$
whose units digit is 6 .
Therefore the units digit of the least positive integer $n$ for which $7 n$ has 2016 digits is 6 .