The string of digits 3691215...300 is formed by merging together the

Question

The string of digits 3691215...300 is formed by merging together the decimal representations of the first 100 positive integer multiples of 3. Counting from the left, what is the 72nd digit of this string of digits?

A. 0
B. 1
C. 6
D. 8
E. 9

A. 0
B. 1
C. 6
D. 8
E. 9

BrentTargetTestPrep · Accepted Answer

1-digit multiples of 3 : 3, 6, 9 (there are 3 numbers for a total of 3 digits)

2-digit multiples of 3 : 12, 15, 18, . . . 99 (there are 30 numbers for a total of 60 digits)

At this point, we have dealt with the first 3 + 60 = 63 digits in the number.

Now we can add the 3-digit multiples of 3 to the end of our 63-digit number:   102  105   108

We see that   8   is the 72nd digit

Bunuel · Answer

Similar questions to practice:

Hope it helps.

Raging · Answer

Thinking about it logically, you can see that there are 33 multiples under 100 with the last multiple being 99. If 33 total multiples are there, then they will have 3 + 30*2 number of digits till 100. That is 63 digits are done. Now you need the 72nd digit which means that you need 3 - 3digit numbers after 99. Which will be 102, 105, 108. Here "8" will be your 72nd digit.

ruis · Answer

Is there a different approach to this question? 
I struggled since the beggining with the "decimal representation" claim... Why does the string of digits end in 300? shouldn´t it end in 99?

Bunuel · Answer

No, the string of digits correctly ends with 300. This is because the sequence consists of the first 100 positive integer multiples of 3. Since 100 times 3 equals 300, the string ends with 300, not 99. The string of digits starts like this:

3 (the first positive integer multiple of 3), 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ..., 297 (the ninety-ninth positive integer multiple of 3), 300 (the hundredth positive integer multiple of 3).

When combined into a continuous string, it looks like:

369121518212427303336...297300.

unraveled · Answer

You can also think of cyclicity of 3 and its multiples here i.e. 3,6,9, 12,15,18 ,21,24,27,30...
have digits 3,6,9...0 repeat for first 10 multiples.
A little thinking tells you that upto multiples of 3 with 2 digit numbers the total digits in the representation are 3 + 60 = 63(as can be seen from posts from members). Now, we are left with next 9 digits(72 - 63).
Therefore, the next there multiples of 3 give the 72nd digit since each of the three multiples are 3 digit number(102,105,108).

Know that the question makes solving it easier as well as difficult if you get the logic. Had it been 71st digit it would have troubled many.

Hence 8.

Answer C.

ektalakra · Answer

think of it like this :
3 *1=3
3*2=6
3*3=9..... 3 digits : which are single digit multiple of 3
from 3*4 till 3*33 ... we have two digit multiples of 3 ..they are (33-4=29 , thus 29+1 = 30 nos that are double digit multiple of 3 .... thus 30*2= 60 digits)
total digits till now are =60+3= 63 digits have been obtained till now. left to obtain 72-63=9 digits
now we need three more 3 digit multiples .
so 3*34 
3*35
3*36= 108 ...where 8 is the 72nd digit.

Username1512 · Answer

The way I was trying to solve it is by calculating when the cycle repeats itself, i.e., after the 19th digit. Then I divided it by 72 and counted to the next 15 digits. Why is that approach wrong?

Bunuel · Answer

Your “cycle” isn’t real because the string doesn’t have a fixed repeating block.

pappal · Answer

just need to check the pattern of first few terms ;  3 6 9   12 15 18   21 24 27  
 30 33 36 39 ...........300
so the followed pattern is 3 digits + 6 digits+6 digits +8 digits   and so on 
up to 99 summation of no. of digits=3+20+20+20=63 (+9=72) means next 3 numbers are enough.
then 102 105 10 8 
so correct answer is 8

Adit_ · Answer

3 singles digit = 3 digits.
30 two-digits = 30*2 = 60 digits.
63 digits covered so far we need 9th digit from here = Last digit of the 3rd number >100
Number = 108, last digit = 8.

Answer:  Option D

The string of digits 3691215...300 is formed by merging together the

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The string of digits 3691215...300 is formed by merging together the

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