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.

Connected Program at Cheenta

Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

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