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12 Days of Christmas GMAT Competition - Day 3: If k is the number form

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buzz111

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18 Dec 2024, 10:41

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Bunuel

12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

If k is the number formed by writing the integers from 1 to 999 sequentially (123456789101112...998999), what is the 341st digit in k?

A. 0
B. 1
C. 4
D. 5
E. 9

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Answer is D or 5

1 to 9 = 9
10 to 99 = 180
100 to 149 = 150
Total 9+180+150 = 339 spaces
next number is 150 which makes 5 the 341 space

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Answer is D. 5

Single digit integers: 1-9 = 9 digits
Double digit integers: 10-99 = 2*90 = 180 digits
Triple digit integers: 100-999 = 3*900 = 2700 digits

All of the single and double digit integers give us 9 + 180 = 189 digits. We need 341 - 189 = 152 more digits, which will all be comprised of triple digit integers now. 152 / 3 = 50R2. We need 50 more integers and two digits after 99, which brings us to 149. The next integer is 150, and the second digit of 150 is 5, which is the 341st digit.

Bunuel

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D

to get to the 341st digit,
first lets count the digits from 1
9(1) + 90(2) + 3 + 3(x) = 341 (9 one digits, 90 two digits, 3 is for 100, and we are looking for the 341st term in the numbers after 100 and before 200)

3x = 149
if x = 50, 3x will become 150.

So then, 9(1)+90(2)+3+3x = 342
the 342nd digit is '0' of '150'
then 341st digit will be '5'.

Lets find the

Bunuel

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UfuomaOh

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what is the 341st digit in k?

We can break the strings of digits from 1-999 into 3 segments

Single digits, double digits and 3 digits

The single digits are 1-9 so the make up 9 numbers and 9 is the 9th number of the array 1-999

10 to 99 make up the 2 digit numbers and there are 180 numbers between 10 to 99

calculated as (99-10+1)*2

this leaves

341-180-9 numbers which is 152 digits

the 341st number is 152 digits between 100 to 999

to know the exact number, we divide 152 by 3

A division would give 50.6

So the exact number is

100+50.6=150.6

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Quote:

If k is the number formed by writing the integers from 1 to 999 sequentially (123456789101112...998999), what is the 341st digit in k?

A. 0
B. 1
C. 4
D. 5
E. 9

1-9 we have 9 digits
10-99 we have 90 numbers and 180 digits

we need 341-189=152 digits more. and 100 onwards we have 3 digits per number.

so 152/3= 50*3 + 2

Hence we are looking for 2nd digit of the 51st number with 3 digits, ie 150 and the answer is 5, ie (D)

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From $1$ to $9$ we have $9$ digits
From $10$ to $99$ we have $90$ numbers each contributing $2$ digits so total of $90*2=180$ digits

Now $341st$ digit is $341-180=152$ digits away

From $100$ to $150$is $51*3=153$ digits

Hence, $0$ is $342nd$ digit and $5$ is $341st$ digit.

IMO Answer is D 5

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Now to solve this question I would count the number of digits from the 1 digit, 2 digit numbers and 3 digit numbers.

1 digit numbers = 1 digit * 9 numbers = 9
2 digit numbers= from 10 to 99 -> total digits = 99-10+1 = 90 numbers * 2 digits = 180 digits
3 digit must be approached differently, we know have 189 digits out of the 341 digits we must conclude so,

341 -189=152 and each number now has 3 digits so to know which number are we on we must divide 152/3=50 with remainder of 2, which is the second digit of the 51 number from 100 and on.

151, its second digit is 5. Correct answer is (D) 5

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Numbers 1-9: 9 single-digit numbers = 9 digits
Numbers 10-99: 90 two-digit numbers = 180 digits (90 x 2)
Numbers 100-999: 900 three-digit numbers = 2700 digits (900 x 3)

first 9 digits cover numbers 1-9.
next 180 digits cover numbers 10-99.
This brings us to digit number 189. We need to find digit number 341, so we still have 341 - 189 = 152 digits to go.

Since we're now in the range of three-digit numbers, divide the remaining digits by 3: 152 / 3 = 50 with a remainder of 2.
This means the 341st digit lies within the 51st three-digit number.
Since the three-digit numbers start at 100, the 51st three-digit number is 150.

The remainder of 2 tells us that the 341st digit is the second digit in the number 150.
the 341st digit in k is 5.

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For writing the first 1-9 integers = 9 digits will required
For the 2 digit numbers (11 to 99) = 2*90 =180 more digits will be required
Already 189 digits are done, for the 341st digit, 341-189 = 152 digits more are required. Since only 3 digit numbers are pending 50 3 digit numbers (3*50) i.e.until 149 would already be done. The next digit 150, of which digit 5 will take the 341st place

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Bunuel

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[color=#0f0f0f]1-9 9 digits
10-99 90#*2=180 digits

total 189
100-109 - 30 digits
So a set of 30*5= 150 digits
(including sets of 100,110,120,130,140 till 149)
total 339 digits so 341st digit will be 150
Hence 5[/color]

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Bunuel

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The digits are written first as 1 to 9
Total number of digits in 1 to 9 = 9
Then, 10 to 99
Total number of digits in 10 to 99 = 2 ( 99 − 10 + 1) = 180
So, from 1 to 99 number of digits = 9 + 180 = 189
We need to find 341st digit,
341 − 189 = 152
3n=152

a1=100, n=152/3=50.67, d=1

an=a1+(n-1)d

an= 100+(50.67- 1)*1

an=149.67

an=150

and 1 should be the answer

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We need to get to the 341st Digit of the consecutive series, now we know the following:

9 numbers from 1-9
90 numbers from 10-99, with 2 digits each, which gives us 180 digits.
Both sum 189 digits we advance in the series. Leaving us 341 - 189= 152 digits left.

These 52 digits are from 3 digit numbers, thus we must divide by 3 to get to the number we are looking for -> 152/30 = 150/3 + 2/3

50 + 2/3 means, we are looking for the 2nd digit (remainder) of the 51st 3-digit number

152 -> Digit (5)

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Let's count the number of digits going into the first 100:
1...9 => $9$ digits
10...90 => $2*90 = 180 $ digits

It means that from three-digit numbers, we need to get $341-189 = 152$ digits
As $152 = 50*3+2$, we see that after 50 three-digit numbers, we will 'land' at the second digit of the remaining fifty-first one.
So, 50 numbers from 100 is 149 => our final number is 150, and the second digit is 5.

The answer is D.

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Bunuel

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[b]is the correct answer.

Digits 1 to 9 will be single-digit numbers (9 * 1 = 9)
Digits 10 to 187 will be double-digit numbers (10-99, i.e, 89 * 2 = 178 and 178 + 9 = 187)
(and the subsequent numbers will be triple digit)

To find the 341st digit in k, 341 - 187 = 154 (distance between the digits)
How many numbers can fit in 154? 154/3 = 51 where 1 is the remainder
100 to 150 (51 numbers) will be present till the 340th digit.
The next number will be "151", so the 341st digit will be "1". [/b]

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From 1-9, there are 9 digits

From 10-19, there are 20 digits, which means from 10 to 99, there are 20*9 = 180 digits

From 100 to 109, there are 30 digits

Since we already have 180 + 9 = 189 digits, consider the numbers till 149, which would give 30*5 = 150 digits

Adding, 150 + 189 = 339

149 = 339 digits

150 = 342 digits

Therefore, the 341st digit is 5

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1 - 9 --> 9 digits
10-99 --> 180 digits ((99-10+1)*2)

341 - 189 = 152

150 is our number and 5 is the 341st digit.

Therefore, D Is the correct answer

Bunuel

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From 1 to 9, we have 9 digits, from 10 to 99 we have 180 digits (20 digits between 10...19 and so on) , from 100 to 199 we have 300 digits (30 digits between 100 and 109 and so on)
The digit we search must have $341-189=152$ position in the line starting with 100 and finish with 199. The line 100...149 contains 150 digits, so the number we are looking for is 150.

1 - 151st digit
5 - 152nd digit (341rd digit)
0 - 153rd digit

Answer: D

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numbers 1-9: 9x1=9 digits
numbers 10-99: 90x2=180 digits

180+9=189
341-189=152

152=50x3+2

51st three-digit number is 150, whose second digit is 5.

IMO D

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Bunuel

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Single Digits: 12...9 => 9 digits
Two digits: 1011...99 => 90*2 => 180 digits
Therefore the number of (1+2) digits => 189 digits
Rest are all 3 digits numbers => 341 - 189 = 152

Now 152/3 = 50.666...
=> 100 101 ... 149 => 50 numbers * 3 => 150 digits
150 => is the next number in line, 2nd digit of 150 => (D) 5 is the 341st digit in k

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Single Digits: 1 to 9 : 9
Double Digits: 10 to 99: 90 (Do remember double digits will take two place, hence 90*2 = 180)
Total places filled till now: 180+9 = 189
Remaining Places: 341-189=152

Three Digits: 100 to 999: But we only need 152 more, so 152/3=50.xx
Next 50 Three digit inter would take 150 places, next no would be 100+50+1=151
so 5 would be the answer.