If the natural numbers starting from 1 are written in sequence (1, 2, 3, 4,…
2025
If the natural numbers starting from 1 are written in sequence (1, 2, 3, 4, ...) without any gaps between them, which digit do you stop at after writing the first 172 digits?
- A.
4
- B.
7
- C.
8
- D.
9
Attempted by 12 students.
Show answer & explanation
Correct answer: D
CONCEPT: When natural numbers are written one after another without gaps, the total digits used up to a point equals the sum of digits contributed by each digit-length range: numbers 1-9 use 1 digit each, numbers 10-99 use 2 digits each, and so on. To find the digit at a given position, add up digits range by range until the position falls inside a range, then locate the exact digit within that number.
Numbers 1 to 9 are 9 single-digit numbers, contributing 9 x 1 = 9 digits.
Digits still needed to reach position 172: 172 - 9 = 163.
From 10 onward every number contributes 2 digits. Dividing the remaining digits by 2: 163 / 2 = 81 remainder 1, so 81 complete two-digit numbers are needed.
The 81st two-digit number starting from 10 is 10 + 81 - 1 = 90. So after writing 10 through 90, the total digit count is 9 + (81 x 2) = 9 + 162 = 171 digits.
The 172nd digit is therefore the very next digit written, which is the first digit of the following number, 91.
The first digit of 91 is 9.
CROSS-CHECK: 171 digits complete the numbers 1 through 90 exactly (9 + 162 = 171), so digit 172 cannot belong to 90 -- it must be the leading digit of 91, confirming the digit is 9.
