If $3^{2019}$ is divided by 10, then what is the remainder?

The correct option is 7.

To find the remainder when $3^{2019}$ is divided by 10, we need to find the units digit of $3^{2019}$ because the remainder of any integer divided by 10 is equal to its last digit (units digit).

Let's observe the pattern of the units digit (cyclicity) of powers of 3:

$3^1=3$ (units digit is 3)
$3^2=9$ (units digit is 9)
$3^3=27$ (units digit is 7)
$3^4=81$ (units digit is 1)
$3^5=243$ (units digit is 3)

Notice that the units digits repeat in a cycle of 4: (3, 9, 7, 1).

To determine where $3^{2019}$ falls in this repeating cycle, we divide the exponent 2019 by 4:

$2019=4\times 504+3$

The remainder when 2019 is divided by 4 is 3.

This means that $3^{2019}$ will have the same units digit as $3^3$, which is 7.

Therefore, when $3^{2019}$ is divided by 10, the remainder is 7.