Select Page

# 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

Number Theory

7/10

##### Suggested Book

Problem Solving Strategies  Excursion In Mathematics

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 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.

# Similar Problems

## Squares and Triangles | AIME I, 2008 | Question 2

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Squares and Triangles.

## Percentage Problem | AIME I, 2008 | Question 1

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Percentage. you may use sequential hints.

## Smallest Positive Integer | PRMO 2019 | Question 14

Try this beautiful problem from the Pre-RMO, 2019 based on Smallest Positive Integer. You may use sequential hints to solve the problem.

## Circles and Triangles | AIME I, 2012 | Question 13

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Circles and triangles.

## Complex Numbers and Triangles | AIME I, 2012 | Question 14

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Complex Numbers and Triangles.

## Triangles and Internal bisectors | PRMO 2019 | Question 10

Try this beautiful problem from the Pre-RMO, 2019 based on Triangles and Internal bisectors. You may use sequential hints to solve the problem.

## Angles in a circle | PRMO-2018 | Problem 80

Try this beautiful problem from PRMO, 2018 based on Angles in a circle. You may use sequential hints to solve the problem.

## Digit Problem from SMO, 2012 | Problem 14

Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2012 based on digit. You may use sequential hints to solve the problem.

## Problem on Semicircle | AMC 8, 2013 | Problem 20

Try this beautiful problem from AMC-8, 2013, (Problem-20) based on area of semi circle.You may use sequential hints to solve the problem.

## Radius of semicircle | AMC-8, 2013 | Problem 23

Try this beautiful problem from Geometry: Radius of semicircle from AMC-8, 2013, Problem-23. You may use sequential hints to solve the problem.