A device can write 100 digits in 1 minute. It starts writing natural numbers. The device is stopped after running it for half an hour. It is found that the last number it was writing is incomplete. The number is :
[amp_mcq option1=”3000″ option2=”3001″ option3=”1026″ option4=”1027″ correct=”option4″]
This question was previously asked in
UPSC CAPF – 2016
– Natural numbers start from 1.
– Digits used for 1-digit numbers (1-9): 9 numbers * 1 digit/number = 9 digits. (Numbers 1 to 9 completed)
– Digits used for 2-digit numbers (10-99): 90 numbers * 2 digits/number = 180 digits. (Numbers 10 to 99 completed)
– Total digits used for 1-digit and 2-digit numbers = 9 + 180 = 189 digits. (Numbers 1 to 99 completed)
– Remaining digits to be written = 3000 – 189 = 2811 digits.
– These remaining digits are used for 3-digit numbers (100-999) and then 4-digit numbers (1000-…).
– Digits used for all 3-digit numbers (100-999): 900 numbers * 3 digits/number = 2700 digits. (Numbers 100 to 999 completed)
– Total digits used for 1-digit, 2-digit, and 3-digit numbers = 189 + 2700 = 2889 digits. (Numbers 1 to 999 completed)
– Remaining digits = 3000 – 2889 = 111 digits.
– These 111 digits are used for writing 4-digit numbers (1000, 1001, …). Each 4-digit number uses 4 digits.
– The digits come from the sequence 1000, 1001, 1002, …
– Number of full 4-digit numbers whose digits are included in the 111 digits = floor(111 / 4) = 27 numbers.
– These 27 numbers are 1000, 1001, …, 1000 + (27 – 1) = 1026.
– Digits used for these 27 full 4-digit numbers = 27 * 4 = 108 digits.
– Total digits used so far = 2889 (up to 999) + 108 (for 1000 to 1026) = 2997 digits.
– The numbers written completely are 1, 2, …, 999, 1000, …, 1026.
– Remaining digits to reach 3000 = 3000 – 2997 = 3 digits.
– These 3 digits are the first three digits of the next number in the sequence, which is 1027.
– The digits of 1027 are 1, 0, 2, 7.
– The device writes the 2998th digit (‘1’ of 1027), the 2999th digit (‘0’ of 1027), and the 3000th digit (‘2’ of 1027).
– The device stops after writing the digit ‘2’ of the number 1027.
– The last number it was writing is 1027, and it is incomplete (only ‘102’ has been written).