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General GMAT Forum Moderator V
Joined: 15 Jan 2018
Posts: 858
Concentration: General Management, Finance
GMAT 1: 720 Q50 V37 N is an 80-digit positive integer in the decimal system. All digits  [#permalink]

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6 00:00

Difficulty:   95% (hard)

Question Stats: 30% (02:32) correct 70% (02:22) wrong based on 40 sessions

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N is an 80-digit positive integer in the decimal system. All digits except for the 44th digit from the left are 2. If N is divisible by 13, find the 44th digit.

A. 1
B. 5
C. 6
D. 8
E. 9

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N is an 80-digit positive integer in the decimal system. All digits  [#permalink]

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2
1
Question:
N is an 80-digit positive integer in the decimal system. All digits except for the 44th digit from the left are 2. If N is divisible by 13, find the 44th digit.

Solution:

Idea to be used: If we have a 6-digit number with all digits identical, say dddddd, where d is a digit, this number is divisible by 7, 11 and 13. Why?
Note: dddddd = ddd x 1000 + ddd = ddd x (1000 + 1)
= ddd x 1001
= d x (111) x (1001)
= d x (3 x 37) x (7 x 11 x 13)

Now, we have 80 digit number where all digits are 2 except the 44th digit from the left.

Let us group six 2s at a time starting from the left (shown below):

222222222222 ...
---6---|---6---|

We will get 7 groups and reach the 42nd digit
Next, we have the 43rd digit which is also 2; and the 44th digit - say d.

After this, there are 36 more digits (all 2s), which can also be grouped six at a time to form 6 groups (see image)

Attachment: 11.JPG [ 16.73 KiB | Viewed 428 times ]

The 7 groups initially and the 6 groups at the end are all divisible by 13

Thus, we just need the number formed by the 42nd and the 43rd digits, i.e. "2x" to be divisible by 13
Thus, x = 6 (since 26 is divisible by 13)

But I doubt this will be asked in the GMAT
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Sujoy Kumar Datta
Director - CUBIX Educational Institute Pvt. Ltd. (https://www.cubixprep.com)
IIT Kharagpur, TU Dresden Germany
GMAT - Q51 & CAT (MBA @ IIM) 99.98 Overall with 99.99 QA
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Intern  B
Joined: 20 Aug 2019
Posts: 44
Re: N is an 80-digit positive integer in the decimal system. All digits  [#permalink]

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sujoykrdatta wrote:
Question:
N is an 80-digit positive integer in the decimal system. All digits except for the 44th digit from the left are 2. If N is divisible by 13, find the 44th digit.

Solution:

Idea to be used: If we have a 6-digit number with all digits identical, say dddddd, where d is a digit, this number is divisible by 7, 11 and 13. Why?
Note: dddddd = ddd x 1000 + ddd = ddd x (1000 + 1)
= ddd x 1001
= d x (111) x (1001)
= d x (3 x 37) x (7 x 11 x 13)

Now, we have 80 digit number where all digits are 2 except the 44th digit from the left.

Let us group six 2s at a time starting from the left (shown below):

222222222222 ...
---6---|---6---|

We will get 7 groups and reach the 42nd digit
Next, we have the 43rd digit which is also 2; and the 44th digit - say d.

After this, there are 36 more digits (all 2s), which can also be grouped six at a time to form 6 groups (see image)

Attachment:
11.JPG

The 7 groups initially and the 6 groups at the end are all divisible by 13

Thus, we just need the number formed by the 42nd and the 43rd digits, i.e. "2x" to be divisible by 13
Thus, x = 6 (since 26 is divisible by 13)

But I doubt this will be asked in the GMAT

Thanks sujoykrdatta for the debrief. May i know what is the range of divisibility rule that is under the GMAT scope?
GMAT Tutor B
Status: Entrepreneur | GMAT, GRE, CAT, SAT, ACT coach & mentor | Founder @CUBIX | Edu-consulting | Content creator
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Re: N is an 80-digit positive integer in the decimal system. All digits  [#permalink]

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Hi.

You should be knowing the divisibility rules for
2, 4, 8 (powers of 2)
5
3 & 9

And also how to tackle non-primes (composites) like 6, 12, 24 etc.

You may check the rule for 11 (though I don't think that will be asked either).

I feel you won't get questions asking about divisibility of 7, 13, etc.

However, you should try to understand the concept of factors of a number like 555555 (the concept I used).

Posted from my mobile device
_________________
Sujoy Kumar Datta
Director - CUBIX Educational Institute Pvt. Ltd. (https://www.cubixprep.com)
IIT Kharagpur, TU Dresden Germany
GMAT - Q51 & CAT (MBA @ IIM) 99.98 Overall with 99.99 QA
_________
Feel free to talk to me about GMAT & GRE | Ask me any question on QA (PS / DS)
Let's converse!
Skype: sk_datta
Alt. Email: sujoy.datta@gmail.com
Director  V
Joined: 27 May 2012
Posts: 971
N is an 80-digit positive integer in the decimal system. All digits  [#permalink]

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sujoykrdatta wrote:
Question:
N is an 80-digit positive integer in the decimal system. All digits except for the 44th digit from the left are 2. If N is divisible by 13, find the 44th digit.

Solution:

Idea to be used: If we have a 6-digit number with all digits identical, say dddddd, where d is a digit, this number is divisible by 7, 11 and 13. Why?
Note: dddddd = ddd x 1000 + ddd = ddd x (1000 + 1)
= ddd x 1001
= d x (111) x (1001)
= d x (3 x 37) x (7 x 11 x 13)

Now, we have 80 digit number where all digits are 2 except the 44th digit from the left.

Let us group six 2s at a time starting from the left (shown below):

222222222222 ...
---6---|---6---|

We will get 7 groups and reach the 42nd digit
Next, we have the 43rd digit which is also 2; and the 44th digit - say d.

After this, there are 36 more digits (all 2s), which can also be grouped six at a time to form 6 groups (see image)

Attachment:
11.JPG

The 7 groups initially and the 6 groups at the end are all divisible by 13

Thus, we just need the number formed by the 42nd and the 43rd digits, i.e. "2x" to be divisible by 13
Thus, x = 6 (since 26 is divisible by 13)

But I doubt this will be asked in the GMAT

Mr sujoykrdatta,
Awesome solution, however there is a small typo, you may wish to correct to prevent confusion. In the figure 2 should be the 43rd digit and the 44th digit should be x. Correspondingly the typo in the last line should be corrected. However the solution is awesome+1.
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- Stne N is an 80-digit positive integer in the decimal system. All digits   [#permalink] 04 Feb 2020, 11:09
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