CAT Quantitative Aptitude Questions | CAT Number Systems - Remainders

CAT Questions | Number Theory | Remainders - Large number

The question is from CAT Number Theory - Remainders. A large number is given and we need to find out the remainder when the first 100 digits is divided by 9. Get practice by solving CAT questions from Remainders.

Question 11: Consider a large number N = 1234567891011121314………979899100. What is the remainder when first 100 digits of N is divided by 9?

  1. 0
  2. 8
  3. 1
  4. 5

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Explanatory Answer

Method of solving this CAT Question from Number Theory - Remainders: Check how many times digits from 1 to 9 occur in first 100 digits of N.

N is nothing but first 100 natural numbers written in ascending order!
The first 99 natural numbers will give us 9 + (90 * 2) =189 digits.
So let us consider first 49 numbers. Number of digits = 9 + (40 * 2) = 89
11 more is needed to make 100 digits. So we can include 50, 51, 52, 53, 54 and 5

Hence the first 100 digits of N goes like this 1234567891011121314.......5253545
We need to find the sum of the above number to check for divisibility by 9.
From 1 to 50, each of the digits 1 to 4 occurs 14 times, digit 5 occurs 6 times and each of 6 to 9 occurs 5 times

Sum of digits from 1 to 4 = 10
Sum of digits from 6 to 9 = 30
Therefore, sum of digits from 1 to 50 = 10 * 15 + 5 * 6 + 30 * 5 = 330
Sum of digits from 51 to 54 plus a 5 = 5 * 4 + ( 1 + 2 + 3 + 4) + 5 = 20 + 15 = 35
Total sum = 330 + 35 = 365
When 365 is divided by 9, we get 5 as the remainder!

The question is "What is the remainder when first 100 digits of N is divided by 9?"

Hence the answer is "5".

Choice D is the correct answer.

 


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