Question

Find the highest power of 2 in $\left[\frac{{10}^{30}}{{10}^{10}+30}\right]$; where [x] is the greatest integer function.

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is B

2

$\left[\frac{{10}^{30}}{{10}^{10}+30}\right]=\left[\frac{{10}^{20}\times {10}^{10}}{{10}^{10}+30}\right]=\left[\frac{{10}^{20}\times {10}^{10}}{{10}^{10}(1+\frac{30}{{10}^{10}})}\right]=\left[\frac{{10}^{20}}{1+\frac{30}{{10}^{10}}}\right]$
Given expression is sum of infinite G.P. (Compare it with $\frac{a}{1-r}$)
$a={10}^{20}$ and $r=(-\frac{3}{{10}^9})$
Given series is ${10}^{20}+{10}^{20}\times (-\frac{3}{{10}^9})+{10}^{20}\times {(-\frac{3}{{10}^9})}^2+{10}^{20}\times {(-\frac{3}{{10}^9})}^3+.........$
${10}^{20}+{10}^{11}\times (-3)+{10}^2\times (-3{)}^2+{10}^{-7}\times (-3{)}^3+.......$ Other terms are very small, we can neglect them.
Only the first three terms need to be considered. In them, the highest power of 2 is 2, as the numbers are a multiple of 4. (Sum of first three terms of given series = 99999999700000000900)