The number (8).(888888........88) is the product of two factors as seen, where the second factor with a larger number of 8's has "K" digits all being 8. Find K if the sum of the digits of the product of the above two numbers is 1000.

Note:- This question is not an original.

The answer is 991.

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First, we can see that: $888x8=7104\\ 8888x8=71104\\ 88888x8=711104\\$

So, to induction I suppose that the product $8..888$ with K chiffres is equal to 711 .. 104, with $(k-2) -1's$ . Then I have that $8x88...8=8x(8x{ 10 }^{ k }+88..8)\\ =6,4x{ 10 }^{ k+1 }+711..104\\ =711...104$

with $(k-2) -1's$ .

So if the sum of digits is 1000, thats mean that $1000=7+4+1(k-2)\\ 989=k-2\\ 991=k$