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.

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