Question

Number of zero's in the product of $5\times 10\times 25\times 40\times 50\times 55\times 65\times 125\times 80$ is :

• A. $8$
• B. $9$
• C. $12$
• D. $13$

Answer 1

The correct option is B

$9$

Solution:

Convert the given expression into prime factors as follows:

$$\begin{gathered} 5\times 10\times 25\times 40\times 50\times 55\times 65\times 125\times 80 \\ =5\times \left(2\times 5\right)\times \left(5^2\right)\times \left(2^3\times 5\right)\times \left(2\times 5^2\right)\times \left(5\times 11\right)\times \left(5\times 13\right)\times \left(5^3\right)\times \left(2^4\times 5\right) \\ =2^9\times 5^{13}\times 11\times 13\left[\mathbf{\because }\mathbf{}{\mathbf{a}}^{\mathbf{m}}\mathbf{\times }{\mathbf{a}}^{\mathbf{n}}\mathbf{=}{\mathbf{a}}^{\mathbf{m}\mathbf{+}\mathbf{n}}\right] \\ =2^9\times 5^9\times 5^4\times 11\times 13 \\ ={\left(2\times 5\right)}^9\times 5^4\times 11\times 13\mathbf{[}\mathbf{\because }\mathbf{}{\mathbf{a}}^{\mathbf{m}}\mathbf{\times }{\mathbf{b}}^{\mathbf{m}}\mathbf{=}{\mathbf{\left(}\mathbf{ab}\mathbf{\right)}}^{\mathbf{m}}\mathbf{]} \\ ={10}^9\times 5^4\times 11\times 13 \end{gathered}$$

Here, the power of 10 is 9.

Therefore, number of Zero's is 9.

Final answer: Hence, option B is the correct answer.