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

A. 32

B. 30

C. 15

D. 28

Solution:

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{aligned} \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{aligned}

[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{aligned} \frac{1 \times - 4 \times - 11 \times - 7}{34} \\ = \frac{28 \times - 11}{34} \\ = \frac{- 6 \times - 11}{34} \\ = \frac{66}{34} \\ R = 32 \end{aligned}

Required remainder = 32

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