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28 Jun 2023, 07:33
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E
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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
28 Jun 2023, 08:04
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rakman123
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
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.
101 103 105 107 109
110th digit = 9
Option E
04 Jul 2023, 06:33
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rakman123
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
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, 109
So the 110th digit is 9.
Answer (E)
