Understand the problem

The product \\((8)(88888……8)\\), where the second factor has k digits, is an integer whose digits have a sum of \\(1000\\). What is k? $\\textbf{(A)}\\ 901\\qquad\\textbf{(B)}\\ 911\\qquad\\textbf{(C)}\\ 919\\qquad\\textbf{(D)}\\ 991\\qquad\\textbf{(E)}\\ 999$

Source of the problem
American Mathematical Contest 10A Year 2014

Topic

Number Theory 

Difficulty Level

7/10

Suggested Book

Problem Solving Strategies  Excursion In Mathematics 

Start with hints

Do you really need a hint? Try it first!

After having a long look into this problem you can first make attempt by listing the first few numbers of the given form.Give it a try!!!!!

So we can do it like this  8*(8)=64 8*(88)=704 8*(888)=7104 8*(8888)=71104 8*(88888)=711104 Now try to observe the pattern in the above table because here lies the main insight of this problem . Come on cook it up!!!!!!    

So form the table you can observe the terms are following a pattern that’s is The first number is 7 Then k-2 number of 1 Then the last two digits are 04 

Now try to make the sum to 1000

So now you are in the final part so you can easily find  7+04+(k-2)=1000

implies 11+(k-2)=1000 . Solving this equation we get the value of K is 991 which is the required answer.

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Math Olympiad Program

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