When positive integer n is divided by 13

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M7MBA: Moderator; Posts: 2058; Joined: Sun Oct 29, 2017 4:24 amFollowed by: 5 members; Thanked: 1 times

by M7MBA » Sat Apr 28, 2018 1:20 am

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When positive integer n is divided by 13, the remainder is 2. When n is divided by 8, the remainder is 5. How many such values are less than 180?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

The OA is B.

How can I solve this PS question? I am confused. Help, please.

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Source: Beat The GMAT — Problem Solving | Sat Apr 28, 2018 1:20 am

Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

Answer

by Vincen » Sat Apr 28, 2018 5:15 am

Hello.

This is the way I solved it. I don't know if it is the best way, but it works. <i class="em em-stuck_out_tongue_winking_eye"></i>

When positive integer n is divided by 13, the remainder is 2 implies that $$n=13\cdot a+2\ ,\ where\ a\in\mathbb{N}.$$ When n is divided by 8, the remainder is 5 implies that $$n=8\cdot b+5\ ,\ where\ b\in\mathbb{N}.$$ Now, let's plug values for a: $$If\ \ a=0\ \ \ then\ \ \ n=2.\ \ Therefore\ b=-\frac{3}{8}\notin\mathbb{N}.\ \ NOT\ POSSIBLE.$$ $$If\ \ a=1\ \ \ then\ \ \ n=15.\ \ Therefore\ b=\frac{10}{8}\notin\mathbb{N}.\ \ NOT\ POSSIBLE.$$ $$If\ \ a=2\ \ \ then\ \ \ n=28.\ \ Therefore\ b=\frac{23}{8}\notin\mathbb{N}.\ \ NOT\ POSSIBLE.$$ $$If\ \ a=3\ \ \ then\ \ \ n=41.\ \ Therefore\ b=\frac{36}{8}\notin\mathbb{N}.\ \ NOT\ POSSIBLE.$$ $$If\ \ a=4\ \ \ then\ \ \ n=54.\ \ Therefore\ b=\frac{49}{8}\notin\mathbb{N}.\ \ NOT\ POSSIBLE.$$ $$If\ \ a=5\ \ \ then\ \ \ n=67.\ \ Therefore\ b=\frac{62}{8}\notin\mathbb{N}.\ \ NOT\ POSSIBLE.$$ $$If\ \ a=6\ \ \ then\ \ \ n=80.\ \ Therefore\ b=\frac{75}{8}\notin\mathbb{N}.\ \ NOT\ POSSIBLE.$$ $$If\ \ a=7\ \ \ then\ \ \ n=93.\ \ Therefore\ b=\frac{88}{8}=11\in\mathbb{N}.\ \ POSSIBLE.$$ $$If\ \ a=8\ \ \ then\ \ \ n=106.\ \ Therefore\ b=\frac{101}{8}\notin\mathbb{N}.\ \ NOT\ POSSIBLE.$$ $$If\ \ a=9\ \ \ then\ \ \ n=119.\ \ Therefore\ b=\frac{114}{8}\notin\mathbb{N}.\ \ NOT\ POSSIBLE.$$ We can continue like this and the next value that is possible is n=197.

Therefore, the unique value that holds the condition and is less than 180 is n=93.

The correct answer is [spoiler]B=1[/spoiler].

I hope it helps.

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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 amFollowed by: 22 members; Location: New York, NY; Thanked: 39 times

by Jeff@TargetTestPrep » Wed May 02, 2018 9:33 am

M7MBA wrote:When positive integer n is divided by 13, the remainder is 2. When n is divided by 8, the remainder is 5. How many such values are less than 180?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

We can create the equation:

n = 13Q + 2

So n can be 2, 15, 28, 41, 54, 67, 80, 93, ...

and

n = 8Q + 5

So n can be 5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93, ...

We see that the first number that satisfies both conditions is 93. To find the other numbers, we can keep adding the LCM of 13 and 8, which is 13 x 8 = 104. Therefore, the next value that satisfies both conditions is 93 + 104 = 197. However, 197 is already greater than 180, so we only have one value, namely 93, that is less than 180 and satisfies both conditions.

Answer: B

Jeffrey Miller
Head of GMAT Instruction
[email protected]

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

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swerve: Legendary Member; Posts: 2535; Joined: Sun Oct 29, 2017 2:04 pmFollowed by: 6 members

by swerve » Wed May 02, 2018 10:50 am

n=13Q1+2
n=8Q2+5

13Q1+2=8Q2+5
13Q1=8Q2+3

When we list the multiples of 13, only when n=91 can we have a remainder of 3 when n is divided by 8.
OR
We will get a multiple of 13 only when Q2=11. Option B.

Regards!

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