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A number N is formed by placing first 100 integers in order

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A number N is formed by placing first 100 integers in order  [#permalink]

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New post 01 Aug 2018, 23:06
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A number N is formed by placing first 100 integers in order (N= 1234567891011121314.......). If N is a 150 digit number, what is the remainder when N is divided by 10?

(A) 1
(B) 3
(C) 7
(D) 8
(E) 9
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Re: A number N is formed by placing first 100 integers in order  [#permalink]

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New post 01 Aug 2018, 23:32
Bunuel, KarishmaB, chetan2u - Please share your approach to the question posted above.
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Re: A number N is formed by placing first 100 integers in order  [#permalink]

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New post 02 Aug 2018, 00:25
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enak wrote:
A number N is formed by placing first 100 integers in order (N= 1234567891011121314.......). If N is a 150 digit number, what is the remainder when N is divided by 10?

(A) 1
(B) 3
(C) 7
(D) 8
(E) 9


What we need is to figure out is the 150th digit of the integer N!

So, if 9 digits(1-9) are consumed by one-digit numbers,
every set of 10 numbers(eg 10-19) will consume 20 more places in N.
Positions 10-29 will be occupied by 10-19
Positions 30-49 will be occupied by 20-29
Positions 50-69 will be occupied by 30-39
Positions 70-89 will be occupied by 40-49
Positions 90-109 will be occupied by 50-59
Positions 110-129 will be occupied by 60-69
Positions 130-149 will be occupied by 70-79

Therefore, the 150th digit is 8(Option D), which is also the remainder when N is divided by 10
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A number N is formed by placing first 100 integers in order  [#permalink]

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New post 02 Aug 2018, 02:07
My approach is as follows:

We have 150 digits, but 100 numbers, and we know that the first 9 numbers take 9 digits, hence 150 - 9 = 141 digits, are left for the remaining two-digit numbers, which start from 10. Since we cannot divide 141 by 2, lets divide 140 by 2 which gives us 70. That means that starting from 10 we will have totally 70 two-digit numbers, hence 150 digit will start with 8 ( remember we stated from 10 => 10 + 70 = 80).

Hope it helps :) !
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Re: A number N is formed by placing first 100 integers in order  [#permalink]

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New post 02 Aug 2018, 02:40
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The pattern is quite simple if closely observed. Every 10th digit repeats itself and then gets incremented by 1.
1 1 2 2 3 3 4 4 5 5 6 6... .
Last digit of number formed from first 10 digits is 1
Last digit of number formed from first 30 digits is 2
...
Hence last digit of number formed from first 150 digits is 8.

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Re: A number N is formed by placing first 100 integers in order  [#permalink]

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New post 02 Aug 2018, 03:08
enak wrote:
A number N is formed by placing first 100 integers in order (N= 1234567891011121314.......). If N is a 150 digit number, what is the remainder when N is divided by 10?

(A) 1
(B) 3
(C) 7
(D) 8
(E) 9


N is obtained by placing integers in order (it shouldn't say first 100 because that defines the number of digits as 192)

N has 150 digits -
The first 9 digits will be 1 - 9. Now we need to figure out the next 141 digits.
Each subsequent number will be a 2 digit number. So we get 140 digits using the next 70 numbers. So these will be integers from 10 to 79. These will give us 140 digits.
The last digit will be the next digit of 80 i.e. it will be 8.

N = 123456789101112....767778798

When divided by 10, the remainder will be the same as the units digit of N i.e. 8.

Answer (D)
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Re: A number N is formed by placing first 100 integers in order  [#permalink]

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New post 02 Aug 2018, 03:20
enak wrote:
A number N is formed by placing first 100 integers in order (N= 1234567891011121314.......). If N is a 150 digit number, what is the remainder when N is divided by 10?

(A) 1
(B) 3
(C) 7
(D) 8
(E) 9


Ans: D

150-9=141

We have 141 digits formed by placing consecutive 2-digit numbers starting from 10.

141/2 = 70.5
=> N ends after placing 70.5 2-digit numbers starting from 10

Placing 70 2-digit numbers starting from 10, we'll have:

123456789101112...777879

(Or 9+70=79)

Placing 0.5 2-digit starting after 79 => placing the first digit of 80 => 8

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Re: A number N is formed by placing first 100 integers in order  [#permalink]

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New post 06 Aug 2018, 06:57
We can answer this question if we know the units digit. To find the units digit of a 150 digits number we follow the following quick strategy:-

First 9 digits + next 140 digits (next 70 two digit integers but the 70th number is 79)

Therefore the 150th digit is 8. Hence the reminder when N is divided by 10 is 8.

Option D is the correct answer.
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Re: A number N is formed by placing first 100 integers in order &nbs [#permalink] 06 Aug 2018, 06:57
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