Question

Let us suppose that: $n = 1234567891011......787980$. The integer $n$ is formed by writing the positive integers in a row, starting with 1 and ending with 80, as shown above. Counting from the left, what is the ${90^{{\text{th}}}}$ digit of $n$?
A. 1
B. 2
C. 3
D. 4
E. 5

Answer 1

Hint: Here, in order to find out the ${90^{{\text{th}}}}$ digit of $n$ we will observe the pattern which is followed and through that pattern the ${90^{{\text{th}}}}$ digit of $n$ will be predicted easily. For analysing such patterns some of the initial digits of n are examined.

Complete step-by-step answer:

Given, $n = 1234567891011......787980$

Here, the first nine digits will be 1 to 9 respectively. The ${10^{{\text{th}}}}$ digit of $n$ will be 1, ${11^{{\text{th}}}}$ digit of $n$ will be 0, ${12^{{\text{th}}}}$ digit of $n$ will be 1 and so on. By this pattern we can say that ${28^{{\text{th}}}}$ digit of $n$ will be 1 and \[{29^{{\text{th}}}}\] digit of $n$ will be 9. Now after this, ${30^{{\text{th}}}}$ digit of $n$ will be 2 and ${31^{{\text{st}}}}$ digit of $n$ will be 0. Similarly, \[{32^{{\text{nd}}}}\] digit of $n$ will be 2 and ${33^{{\text{rd}}}}$ digit of $n$ will be 1.

According to this pattern, ${48^{{\text{th}}}}$ digit of $n$ will be 2 and ${49^{{\text{th}}}}$ digit of $n$ will be 9. Now after this, ${50^{{\text{th}}}}$ digit of $n$ will be 3 and ${51^{{\text{st}}}}$ digit of $n$ will be 0. Similarly, ${52^{{\text{nd}}}}$ digit of $n$ will be 3 and ${53^{{\text{rd}}}}$ digit of $n$ will be 1. According to this pattern, ${68^{{\text{th}}}}$ digit of $n$ will be 3 and ${69^{{\text{th}}}}$ digit of $n$ will be 9.

Now after this, ${70^{{\text{th}}}}$ digit of $n$ will be 4 and ${71^{{\text{st}}}}$ digit of $n$ will be 0. Similarly, ${72^{{\text{nd}}}}$ digit of $n$ will be 4 and ${73^{{\text{rd}}}}$ digit of $n$ will be 1. According to this pattern, ${88^{{\text{th}}}}$ digit of $n$ will be 4 and ${89^{{\text{th}}}}$ digit of $n$ will be 9. Now after this, ${90^{{\text{th}}}}$ digit of $n$ will be 5.

So, the ${90^{{\text{th}}}}$ digit of $n$ will be 5 (${90^{{\text{th}}}}$ digit and ${91^{{\text{st}}}}$digit together makes the positive integer 50).

Therefore, option E is correct.

Note: In these types of problems, there always exists a pattern or sequence and we just have to analyse that pattern to reach the answer. In this case, patterns exist for 1 to 9 positive integers, 10 to 19, 20 to 29, 30 to 39 and so on.