Sum of digits To the Infinity : ( 2 )

Algebra Level pending

Find the smallest value of $X$ such that sum of digits of the decimal representation of $1 + 2 + 3 + \cdots + (X \cdot 10^n)$ is equal to 3 for any positive integer $n$ .

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1 solution

محمد أبو العمايم
Mar 2, 2021

Let : S = 1 + 2 + 3 + . . . . . . . . + X . 10^n ; so we Note that : S = [ X . ( 10^n ) / 2 ] + [ X^2 . ( 10^2n ) / 2 ] .. So Sum of All digits of ( S ) = ( X / 2 ) + ( X^2 ) / 2 = 3 ; X + X^2 = 6 ; ( X^2 ) + X - 6 = 0 ; ( X - 2 )( X + 3 ) = 0 ; X = 2 .

Of course, $X=20$ , $X=200$ , etc also work ( $X$ isn't unique here)

Chris Lewis - 3 months, 1 week ago

Thanks. I've clarified that we're looking for the smallest value of $X$ .

In the future, if you have concerns about a problem's wording/clarity/etc., you can report the problem. See how here .

Brilliant Mathematics Staff - 3 months, 1 week ago

X = 20 OR 200 OR 2000 ; etc will Make : X . 10^n : Thanks .

محمد أبو العمايم - 3 months, 1 week ago

Sum of digits To the Infinity : ( 2 )

1 solution

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