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

Explanation:-

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.]

$$\frac{{5 \times 7 \times 10 \times 23 \times 27 \times 3}}{{34}}$$      ⇒ $$\frac{{35 \times 30 \times 23 \times 27}}{{34}}$$    [Number Multiplied]
$$\frac{{35 \times 30 \times 23 \times 27}}{{34}}$$     ⇒ $$\frac{{1 \times - 4 \times - 11 \times - 7}}{{34}}$$

[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.]

$$\frac{{1 \times - 4 \times - 11 \times - 7}}{{34}}$$     ⇒ $$\frac{{28 \times - 11}}{{34}}$$   ⇒ $$\frac{{ - 6 \times - 11}}{{34}}$$   ⇒ $$\frac{{66}}{{34}}$$  ⇒ $${\text{R}}$$ ⇒ 32
Required remainder = 32.