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# AMC 10A Year 2014 Problem 20 Sequential Hints

A challenging number theory problem. Here the main idea is the visualization of a pattern of which appeared in the multiplication.

# Understand the problem

[/et_pb_text][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway||||||||” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|15px||” box_shadow_style=”preset2″]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? $\\textbf{(A)}\\ 901\\qquad\\textbf{(B)}\\ 911\\qquad\\textbf{(C)}\\ 919\\qquad\\textbf{(D)}\\ 991\\qquad\\textbf{(E)}\\ 999$

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”3.25″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_accordion open_toggle_text_color=”#0c71c3″ _builder_version=”3.27″ toggle_font=”||||||||” body_font=”Raleway||||||||” text_orientation=”center” custom_margin=”10px||10px”][et_pb_accordion_item title=”Source of the problem” open=”on” _builder_version=”3.27″]American Mathematical Contest 10A Year 2014

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Number Theory

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7/10

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Problem Solving Strategies  Excursion In Mathematics

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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!!!!!

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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!!!!!!

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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

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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.