Question

If ${10^{\rm{x}}} - 2017$ is expressed as integer. What is the sum of its digits?

Answer 1

Hint:
Try to make a general formula by considering the smaller powers of 10 first. Then proceed with higher powers of ten and try to make a general formula.

Complete step by step solution:
Here we will consider a general case
${10^{\rm{x}}} - 2017$
Taking x = 4, ${10^4} - 2017 = 10000 - 2017 = 7983$
Taking x = 5, ${10^5} - 2017 = 100000 - 2017 = 97983$
Similarly
x = 6, ${10^6} - 2017 = 1000000 - 2017 = 997983$
x = 7, ${10^7} - 2017 = 10000000 - 2017 = 9997983$ and so on.
Here we note that if the value of x is greater than 1 the last four digits of ${10^{\rm{x}}} - 2017$ are always 7, 9, 8 and 3.
So, from here we can make a formula for the sum of digits which is
The sum of digits $ = \left( {{\rm{x}} - 4} \right)\left( 9 \right) + 7 + 9 + 8 + 3$ where x is power of 10.
Hence, for ${10^{\rm{x}}} - 2017$ ,
Sum of digits $ = \left( {2017 - 4} \right)\left( 9 \right) + 7 + 9 + 8 + 3{\rm{\;\;}}$
 $ = 2013\left( 9 \right) + 7 + 9 + 8 + 3$
 $ = 18117 + 7 + 8 + 3$
=18,144

Therefore, the sum of its digit is 18,144

Note:
Here you should note that this formula applied only for when x is greater than 4 because ${10^x}$ where x is less than 4, will be smaller than 2017 which will result in a negative integer.