JUST A MODULO PROBLEM!!(3)

Number Theory Level 3

The number 1234567891011..........979899100 when divided by 9 leaves a remainder "X". Find (9-x).


The answer is 8.

Jaiveer Shekhawat by jaiveer shekhawat , 22, India

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2 solutions

Jaiveer Shekhawat
Sep 18, 2014

THERE'S SOMETHING SPECIAL ABOUT THE NO. "9":-

we know,

1=1(mod9)

12=3(mod9)

123=6(mod9)

1234=1(mod9)

12345=6(mod9)

123456=12=3(mod9) [add until you get a one digit no.]

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Did you notice something??

The remainder will be the remainder obtained by dividing the sum of the digits of the number by 9.

So, the sum of our number will be...

1+2+3+.....+99+100=5050=10=1

Thus,

1234567891011..........979899100=1(mod 9)

Thus, x=1. Then, 9-x =8.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Another you could do is to add all the digits in that number which will be 901.Now if you divide the sum by 9 you will get the remainder when that number is divided by 9,which is 1.

Anik Mandal - 6 years, 8 months ago
Sophie Crane
Sep 27, 2014

A number is congruent to the sum of its digits modulo 9. The number 12345....979899 contains each digit from 1 to 9 exactly 18 times, and we don't care about the 0s. Thus 12345...979899 is divisible by 9. So 12345...979899100 is congruent to 1 mod 9. Thus x=1 and 9-x=8

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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