Find the remainder when 73 × 75 × 78 × 57 × 197 × 37 is divided by 34.

Answer: Option A

Remainder,
$$\frac{73\times 75\times 78\times 57\times 197\times 37}{34}$$
$$=\frac{5\times 7\times 10\times 23\times 27\times 3}{34}$$
[We have taken individual remainder, which means if 73 is divided by 34 individually, it will give remainder 5, 75 divided 34 gives remainder 7 and so on.]

$$\begin{array}{rl}& \frac{5\times 7\times 10\times 23\times 27\times 3}{34}\\ & =\frac{35\times 30\times 23\times 27}{34}\\ & =\frac{1\times -4\times -11\times -7}{34}\end{array}$$

[We have taken here negative as well as positive remainder at the same time. When 30 divided by 34 it will give either positive remainder 30 or negative remainder -4. We can use any one of negative or positive remainder at any time.]

$$\begin{array}{rl}& \frac{1\times -4\times -11\times -7}{34}\\ & =\frac{28\times -11}{34}\\ & =\frac{-6\times -11}{34}\\ & =\frac{66}{34}\\ & \text{R}=32\end{array}$$
Required remainder = 32