If 56 × 75 × 60 × 84 × 210 = 2^p × 3^q × 5^r × 7^s, then what is the value of $$\left[ {\frac{{\left( {{\text{p}} + {\text{q}}} \right)}}{{\text{s}}}} \right] + {\text{r}}?$$

A. 6

B. 8

C. 12

D. 10

Answer: Option B

Solution(By Examveda Team)

56 × 75 × 60 × 84 × 210 = 2^p × 3^q × 5^r × 7^s
7 × 2³ × 5² × 3 × 2² × 3 × 5 × 2² × 3 × 7 × 7 × 3 × 2 × 5 = 2^p × 3^q × 5^r × 7^s
2⁸ × 3⁴ × 5⁴ × 7³ = 2^p × 3^q × 5^r × 7^s
∴ p = 8
q = 4
r = 4
s = 3
$$\eqalign{ & \therefore \left[ {\frac{{\left( {{\text{p}} + {\text{q}}} \right)}}{{\text{s}}}} \right] + {\text{r}} \cr & = \left[ {\frac{{\left( {8 + 4} \right)}}{3}} \right] + 4 \cr & = \frac{{12}}{3} + 4 \cr & = 4 + 4 \cr & = 8 \cr} $$