, 2012

The units digit of any product is given by the units digit of the product of the units digits of the numbers being multiplied.

For example, the units digit of the product $12\times53$ is given by the product $2\times3$, so it is 6.

Thus to determine the units digit of $N$, we need only consider the product of the units digits of the numbers being multiplied to give $N$.

The units digits of the numbers in the product $N$ are $1,3,7,9,1,3,7,9,\dots$, and so on.

That is, the units digits $1,3,7,9$ are repeated in each group of four numbers in the product.

There are ten groups of these four numbers, $1,3,7,9$, in the product.

We first determine the units digit of the product $1\times3\times7\times9$.

The units digit of $1\times3$ is 3.

The units digit of the product $3\times7$ is 1 (since $3\times7=21$).

The units digit of $1\times9$ is $9$.

Therefore, the units digit of the product $1\times3\times7\times9$ is 9.

(We could have calculated the product $1\times3\times7\times9=189$ to determine the units digit.)

This digit 9 is the units digits of the product of each group of four successive numbers in $N$.

Thus, to determine the units digit of $N$ we must determine the units digit of

$9\times9\times9\times9\times9\times9\times9\times9\times9\times9$.

This product is equal to $81\times81\times81\times81\times81$.
Since we are multiplying numbers with units digit 1, then the units digit of the product is 1.