Question

What will be the value of x for $\frac{({100}^{17}-1)+({10}^{34}+x)}{9}$; the remainder =0?

• A. 3
• B. 6
• C. 9
• D. 8

Answer 1

The correct option is D 8

$\frac{({100}^{17}-1)+({10}^{34}+x)}{9}{100}^{17}-1=\frac{\mathrm{1000...00}-1}{17zerores}=\frac{\mathrm{9999...99}}{16nines}\Rightarrow$ divisible by $9\Rightarrow R=0$
Since the first part of the expression is giving a remainder of 0, the second part should also give 0 as a remainder if the entire remainder of the expression has to be 0. Hence, we now evaluate the second part of the numerator.
${10}^{34}+x=\frac{\mathrm{1000..00}+x}{34zerores}=\frac{\mathrm{1000...00}x}{33zeroes}$
with x at the right most place. In order for this number to be divisible by 9, the sum of digits should be divisible by 9.
$\Rightarrow 1+0+\mathrm{0...}+0+x$ should be divisible by 9.
$\Rightarrow 1+x$ should be divisible by $9\Rightarrow x=8$
$\Rightarrow$ Option (d)