Explanation:

Step 1: Simplify the expression
Given expression: $(3)^{1989}$
To find the remainder when $(3)^{1989}$ is divided by 7, we can simplify the expression using the concept of modular arithmetic.

Step 2: Apply modulo operation
We know that $(a \times b) \mod n = ((a \mod n) \times (b \mod n)) \mod n$
Therefore, we can apply the modulo operation to simplify the expression.
$(3)^{1989} \mod 7 = ((3 \mod 7)^{1989}) \mod 7$
Now, we need to find the remainder when 3 is divided by 7.
3 mod 7 = 3

Step 3: Find the remainder
Now, we have $(3)^{1989} \mod 7 = (3^{1989}) \mod 7$
To find the remainder when $3^{1989}$ is divided by 7, we can use the concept of Euler's theorem or Fermat's little theorem.
Euler's theorem states that if a and n are relatively prime, then $a^{\phi(n)} \equiv 1 \mod n$
Since 3 and 7 are relatively prime, we can use Euler's theorem to simplify the expression.
$\phi(7) = 7 - 1 = 6$
Therefore, $3^{1989} \mod 7 = 3^{1989 \mod 6} \mod 7$
$1989 \mod 6 = 3$
So, $3^{1989} \mod 7 = 3^3 \mod 7 = 27 \mod 7 = 6$

Step 4: Conclusion
Therefore, the remainder when $(3)^{1989}$ is divided by 7 is 6.