Computer Science problem by Giorgio Coniglio

Number Theory Level pending

Find the smallest integer $n$ such that $2^n$ starts with 10 nines .

$2^n = \underbrace{\overline{9999999999......abcdefghij}}_{\text{A few hundred million digits}}$

What is the sum of last 10 digits of the smallest number $2^n$ , namely $a+b+c+d+e+f+g+h+i+j =?$

Note : There are infinitely many n such that $2^n$ starts with 10 nines. We are looking for the smallest such n .

1 solution

Giorgio Coniglio
Oct 25, 2016

2^n = 9999999999 ........ abcdefghij is a huge number with n = 1,923,400,330 so the number is:

2^1,923,400,330 = 99999999997213828437359271518597759670596401080398033 <<579001087>> 66065043081649971975212712782767237949748583359053824

and the last 10 digits are: 3 3 5 9 0 5 3 8 2 4 and their sum is 42.

The solution to the last part after finding n is easy using modulus arithmetic 2^1,923,400,330 mod 10,000,000,000 to get the last 10 digits.

Are you absolutely certain that $n = 1923400330$ satisfies the conditions?

If yes, how did you perform the calculation / verification?

Calvin Lin Staff - 4 years, 7 months ago