CAT Practice : Number System - Remainders

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Check how many times digits from 1 to 9 occur in first 100 digits of N.

Remainders - Large Number

    Q.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

 

  • Correct Answer
    Choice D. 5

Detailed Solution

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!

Correct Answer: 5

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When we divide 24 pigeons into 5 groups, 4 are left out or remain. This is the remainder. When there is a question based on this idea in the exam, we have a CAT among the pigeons.