remainder problem

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fangtray: Master | Next Rank: 500 Posts; Posts: 273; Joined: Thu Sep 08, 2011 6:50 amFollowed by: 3 members; Thanked: 5 times

by fangtray » Thu May 10, 2012 3:01 pm

When the even integer n is dividd by 9, the remainder is 8. Which of the following when added to n, gives a sum that is divisible by 18?

a. 1
b. 4
c. 5
d. 10
e. 17

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Source: Beat The GMAT — Problem Solving | Thu May 10, 2012 3:01 pm

vatish: Newbie | Next Rank: 10 Posts; Posts: 4; Joined: Mon May 07, 2012 4:08 pm

by vatish » Thu May 10, 2012 3:56 pm

N = 9a + 8
As N is even 9a has to be even or a has to be even. For a = 2, N = 26
Now check answer choices.

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gmatNooB8787: Senior | Next Rank: 100 Posts; Posts: 31; Joined: Sun Oct 16, 2011 10:02 amFollowed by: 3 members; Thanked: 4 times

by gmatNooB8787 » Thu May 10, 2012 4:59 pm

We know that and q is even so. q/2 = integer

n = q9 + (8+x) ;

n/18 = 9*q / 18 + (8+x)/18;
= q/2 + (8+x)/18;
= (integer) + Integer

when will (8+x) /18 be an integer i.e. for what value of x ?

Ans : D i.e. 10

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

by Stuart@KaplanGMAT » Thu May 10, 2012 10:05 pm

fangtray wrote:When the even integer n is dividd by 9, the remainder is 8. Which of the following when added to n, gives a sum that is divisible by 18?

a. 1
b. 4
c. 5
d. 10
e. 17

Hi!

Picking numbers is a great approach to solving many number property questions.

Here, our constraints are that n/9 must have a remainder of 8 and n must be even. What's the smallest number that satisfies these constraints? n=8 (8/9 has a quotient of 0 and a remainder of 8).

Now let's find the answer that, when we add it to 8, gives us a multiple of 18.

a) 1... 9 is not a multiple of 18... out!
b) 4... 12 is not a multiple of 18... out!
c) 5... 13 is not a multiple of 18... out!
d) 10... 18 IS a multiple of 18... ding ding ding we have a winner!

Of course, we could have also just applied a bit of logic. Since the answers are all pretty small, we can just ask "what number when added to 8 gives us 18, the first multiple of 18?" - and of course (d) would have jumped right out as correct.

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LalaB: Master | Next Rank: 500 Posts; Posts: 425; Joined: Wed Dec 08, 2010 9:00 amFollowed by: 7 members; Thanked: 56 times; GMAT Score:690

by LalaB » Fri May 11, 2012 11:00 am

n=2k

2k=9x+8
if x=1 then n=9+8=17 (out, since n is even)
if x=2 then 9*2+8=26 (bingo! n is even)

now just pay attention that 26 is 10 away from 36, which is divisible by 18

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mcdesty: Master | Next Rank: 500 Posts; Posts: 116; Joined: Wed Mar 14, 2012 1:02 pmFollowed by: 11 members; Thanked: 20 times; GMAT Score:760

by mcdesty » Thu Aug 21, 2014 10:07 am

Here is how I would tackle this one.

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

by GMATinsight » Thu Aug 21, 2014 11:10 am

fangtray wrote:When the even integer n is divided by 9, the remainder is 8. Which of the following when added to n, gives a sum that is divisible by 18?

a. 1
b. 4
c. 5
d. 10
e. 17

Given : When the even integer n is divided by 9, the remainder is 8
i.e. n could be 9x2+8 or 9x4+8 and so on...
i.e. n could be 26 or 44 and so on...

Question : Which of the following when added to n, gives a sum that is divisible by 18

Result will be divisible by 18 only When it's an Even integer therefore n must be added with an even integer only ELIMINATING OPTIONS A, C, and E

Checking Option B
26+4 = 18 NOT Divisible by 18 Ruled out
Therefore Option D must be correct

Checking option D
26+10 = 36 Divisible by 18
44+10 = 54 Divisible by 18

Answer: Option D

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Thu Aug 21, 2014 2:42 pm

Here's a general approach to any problem like this.

Consider a few integers with a remainder of 8 when divided by 9:

9*1 + 8 = 17
9*2 + 8 = 26
9*3 + 8 = 35
9*4 + 8 = 44
9*5 + 8 = 53
9*6 + 8 = 62
etc.

Notice how all the even ones are of the form 18*(something) + 8?

That means n = 18k + 8, where k is some integer. (For instance, if n = 26, k = 1, and if n = 44, k = 2.)

For this to divide by 18, we need to get rid of the remainder (which is currently 8). So if we add any number with a remainder of 10 when divided by 8, we'll get ...

(18k + 8) + (18m + 10) = 18k + 18m + (10+8) = 18k + 18m + 18, which must divide by 18.

For instance, suppose we have 26 + 28. This gives us 18 + 8 + 18 + 10, or 18 + 18 + 18. Success!

Similarly, ANY answer of the form 18*something + 10 works. Since 10 = 18*0 + 10, the answer must be D.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Aug 21, 2014 3:42 pm

fangtray wrote:When the even integer n is dividd by 9, the remainder is 8. Which of the following when added to n, gives a sum that is divisible by 18?

a. 1
b. 4
c. 5
d. 10
e. 17

When the even integer n is divided by 9, the remainder is 8.
In other words, n is a MULTIPLE OF 9 plus 8:
n = 9a + 8, where aâ‰¥0.

Options for n, if n can be even or odd:
n = 8, 17, 26, 35...

Since n must be even, only the values in red are viable:
n = 8, 26...

When the correct answer choice is added to 8, the result must be a multiple of 18.
Only D works:
10 + 8 = 18.

The correct answer is D.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Fri Aug 22, 2014 10:43 am

fangtray wrote:When the even integer n is divided by 9, the remainder is 8. Which of the following when added to n, gives a sum that is divisible by 18?

a. 1
b. 4
c. 5
d. 10
e. 17

There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2

When the even integer n is divided by 9, the remainder is 8
In other words, when n is divided by 9 we get some integer (say k) with remainder 8
So, applying the above rule, we can write n = 9k + 8 (where k is an integer)

IMPORTANT: We're told that n is EVEN.
In other words, 9k + 8 is even
Since 8 is even, we can conclude that 9k is even, which means k is EVEN.
If k is even, then 9k = 9(2 times some integer) = (18)(some integer)
So, we can conclude that 9k is divisible by 18

n = 9k + 8
We can see that, if we add 10 to n, we get 9k + 8 + 10, which simplifies to 9k + 18
Since 9k is divisible by 18 AND 18 is divisible by 18, we know that 9k + 18 is divisible by 18

So, if we add 10 to n, the sum is divisible by 18

Answer: D

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Katy_: Junior | Next Rank: 30 Posts; Posts: 22; Joined: Sun Aug 17, 2014 6:53 am; Thanked: 1 times

by Katy_ » Sun Aug 24, 2014 4:58 am

fangtray wrote:When the even integer n is dividd by 9, the remainder is 8. Which of the following when added to n, gives a sum that is divisible by 18?

a. 1
b. 4
c. 5
d. 10
e. 17

Could you please comment on my solution?

n = 9m+8 => (n+1) is divisible by 9

=> (n+1 + 9k) is divisible by 9

n is divisible by 18 when (n+1+9k) is an even number (or divisible by 2)

n is even => n+1 is odd

so 9k must be an odd number (odd+odd=even)

=> k must be odd

We have a lot of k but within this question's options, we just need to care about 1 values of k:

k=1 we have 1+9k = 10.

So 10 is the answer.

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