The string of digits 135791113...999 is formed by

Question

The string of digits 135791113...999 is formed by merging together the decimal representations of the odd integers from 1 through 999. Counting from left, what is the 110th digit of this string of digits?

A) 0
B) 3
C) 5
D) 7
E) 9

A) 0
B) 3
C) 5
D) 7
E) 9

gmatophobia · Accepted Answer

A good one!

Let's visualize this

\quad1 \quad3 \quad5 \quad7 \quad9 \quad11 \quad13 \quad....\quad 997 \quad 999

This is an arithmetic sequence with a common difference = 2

We want to find the 110th digit of this string (counting from the left)

Length of the string contributed by the 1 digit number = 5 (i.e. 13579)

Each of the two-digit numbers will contribute a length of two to the string

Total length contributed by the two-digit numbers = Number of two-digit numbers * 2

Number of two-digit numbers = \frac{ (99 - 11) }{ 2 }+ 1 = 45

length contributed by the two-digit numbers = 45 * 2 = 90

Total Length obtained so far = 90 + 5 = 95

Remaining length = 110 - 95 = 15

Each three-digit number will contribute a length of 3 to the string. Hence, we need  \frac{15}{3} = 5  three digit numbers.

101 103 105 107 10  9

110th digit = 9

Option E

KarishmaB · Answer

Talking about positive integers, first we have 9 one digit integers (from 1 to 9), then 90 two digit integers (from 10 to 99) and then 900 three digit integers.

If you are not sure how to calculate, say how many two digit integers we have, you know that we have 10 to 99 two digit integers. Number of integers in this case is 99 - 10 + 1 = 90
(We take + 1 because 10 is also a two digit integer)

Of the 9 one digit integers, 5 are odd because we start with 1 and end with 9 (both odd).
Of the 90 two digit integers, 45 are odd because we start with 10 and end with 99 (one even one odd so equal split)

This accounts for total 5 + 2 * 45 = 95 digits.

We still have another 15 digits to go. Now all numbers will be three digit numbers. I simply need the next 5 three digit numbers. 
101, 103, 105, 107, 10 9

So the 110th digit is 9.

Answer (E)

Nabneet · Answer

Let's break down the pattern into sections:
1, 3, 5, 7, 9, 11, 13, ..., 99, 101, 103, ..., 999

The first section consists of the single-digit odd integers: 1, 3, 5, 7, 9. Each digit appears once, so the total number of digits in this section is 5.

The second section consists of the two-digit odd integers: 11, 13, ..., 99. There are 45 two-digit odd integers, and each number has two digits, so the total number of digits in this section is 45 * 2 = 90.

Therefore, the first 95 digits of the string come from the first two sections: 5 digits from the single-digit section and 90 digits from the two-digit section.

13579.....99 (total 95 digits) -----

13579.....99 next section takes 3 places so (101...109) adds up 15 places

13579.....99101103105....10 9

9 is the 110th digit in the sequence

Archit3110 · Answer

single digit odd will be 5
2 digit odd will be 11 to 99 ; 2*5*9 ; 90
95 digit string from 1 to 99
110-95 ; 15
101 103 105 107 10 9  ; 9 will be the 105th digit from left
option E

JeffTargetTestPrep · Answer

From 1 to 99 there are 50 odd integers, consisting of 5 single-digit and 45 two-digit odd integers.

So far, that’s a total of 5 + 45 × 2 =95 digits.

We still need 110 – 95 = 15 digits, which can be exactly provided by the first five (=15/3) three-digit odd integers.

The fifth three-digit odd integer is 109, whose last digit is 9.

Therefore, the 110th digit in question is 9.

Answer: E

wtstone · Answer

Can anyone explain what the meaning of "decimal representations" in this question? Seems these numbers are represented as integers being combined to form a continuous string.

GMATGuruNY · Answer

An alternate approach is to write out the digits in ROWS OF 10 and look for a pattern:

1 3 5 7 9 1 1 1    3 1   
5 1 7 1 9 2 1 2    3 2   
5 2 7 2 9 3 1 3    3 3

As illustrated by the digits in blue, the row endings exhibit the following pattern:
1st row --> 31 (where 1 is the tens digit for 15)
2nd row --> 32 (where 2 is the tens digit for 25)
3rd row --> 33 (where the second 3 is the tens digit for 35)
Implication:
9th row --> 39 (where 9 is the tens digit for 95)

The 9th 10-digit row brings the total number of digits to 90.
New two rows, beginning with the units digit for 95:
5 9 7 9 9 1 0 1 1 0
3 1 0 5 1 0 7 1 0    9

The digit in green is the 110th in the string.

E

Bunuel · Answer

Base-10 notation, also known as decimal notation, is a method of representing numbers using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This is the conventional system for denoting numbers, in contrast to systems like the binary numeral system, which is a base-2 number system.

So, the odd integers from 1 through 999, written as the string of digits would simply be odd numbers written one after another: 135791113...999

epicstudent · Answer

I attached my work to this problem. Hope it is helpful!

Bunuel · Answer

Similar questions to practice:

Hope it helps.

Apk641 · Answer

1. One digit numbers length. 1-9=5

2. Two digit numbers length
99-11/2+1= 45 numbers
45*2=90

3. Total length so far for 1-99 digits= 5+90=95
110-95=15 left

If we count from 99-- 101 103 105 107 109
9 gets us to digit 110

Answer is 9

Dbrunik · Answer

Everybody here is writing explanations that require formulas and extra things to memorize. 
just think about it logically

1,3,5,7,9 is 5 numbers 
11,13,15,17,19, is 10 number

99 will bring you to 90 numbers. 90+5=95. 99 is the 95th number

101,103,105,107,10 9  = 15 numbers. 95+15=110. the final number is 9.

saurvan · Answer

ITs a little unclear. I solved this by counting 1-9 as 5 contributions, 11-19 as 10 contributions bcas its a fused string, so "11" is 2 numbers IMO. I am getting the interpretation wrong at times sigh :/

GMATinsight · Answer

The number of single digit number use = 5 {1, 3, 5, 7, 9}

Total Digit count = 5

Number of ODD two-digit numbers from 1 to 100 = 45

Digit count = 45*2 = 90

Total Digit count so far from 1 to 99 = 5+90 = 95

i.e. Now we just have to reach 15th digit from here to get 115th digit

All numbers from hereon are three-digit numbers (till we get answer)

so 5th 3-digit number's unit digit is our answer

{101, 103, 105, 107, 10  9  }

115th Digit = 9

Answer: Option E

antonis22800 · Answer

Let me know if my method is just a coincidence please.

We notice the difference is 2. Also we notice the following pattern: 
3rd digit of the string is 2*3=6 - 1 = 5. 
5th digit of the sting is 2*5 = 10 - 1 = 9.

Thus, 
110th is 2*110 = 220 - 1 =219 so answer choice E.

Saksham23 · Answer

Not sure if my reasoning is correct, but if the pattern is odd integers, and we have to ideally find the 110th digit, we can use an = a+ (n-1)*d where, a = 1, n = 110, d = 2. 
This gives me an = 219, so the 110th digit is 9

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