GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 22 Oct 2019, 16:58

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# If w, x, y, and z are the digits of the four-digit number N, a positiv

Author Message
TAGS:

### Hide Tags

Senior Manager
Joined: 18 Aug 2009
Posts: 250
If w, x, y, and z are the digits of the four-digit number N, a positiv  [#permalink]

### Show Tags

Updated on: 07 Aug 2019, 03:12
7
28
00:00

Difficulty:

55% (hard)

Question Stats:

59% (01:30) correct 41% (01:41) wrong based on 599 sessions

### HideShow timer Statistics

If w, x, y, and z are the digits of the four-digit number N, a positive integer, what is the remainder when N is divided by 9?

(1) w + x + y + z = 13
(2) N + 5 is divisible by 9

Originally posted by gmattokyo on 08 Nov 2009, 03:43.
Last edited by Bunuel on 07 Aug 2019, 03:12, edited 2 times in total.
Math Expert
Joined: 02 Sep 2009
Posts: 58427
Re: If w, x, y, and z are the digits of the four-digit number N, a positiv  [#permalink]

### Show Tags

08 Nov 2009, 05:17
13
1
19
gmattokyo wrote:
If w, x, y, and z are the digits of the four-digit number N, a positive integer, what is the remainder when N is divided by 9?
1. w + x + y + z = 13
2. N + 5 is divisible by 9

Remainder when a number is divided by 9 is the same as remainder when the sum of its digits is divided by 9:

$$Remainder \frac{N}{9}=Remainder \frac{w + x + y + z}{9}$$

Let's show this on our example:

Our 4 digit number is $$1000w+100x+10y+z$$. what is the remainder when it's divided by 9?

When 1000w is divided by 9 the remainder is $$\frac{w}{9}$$:

$$\frac{3000}{9}$$ remainder $$3$$, $$\frac{3}{9}$$remainder $$3$$.

The same with $$100x$$ and $$10y$$.

So, the remainder when $$1000w+100x+10y+z$$ is divided by 9 would be:

$$\frac{w}{9}+\frac{x}{9}+\frac{y}{9}+\frac{z}{9}=\frac{w+x+y+z}{9}$$

(1) w + x + y + z = 13 --> remainder 13/9=4, remainder N/9=4. Sufficient.

(2) N+5 is divisible by 9 --> N+5=9k --> N=9k-5=4, 13, 22, ... --> remainder upon dividing this numbers by 9 is 4. Sufficient.

_________________
##### General Discussion
CEO
Joined: 17 Nov 2007
Posts: 3041
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Re: If w, x, y, and z are the digits of the four-digit number N, a positiv  [#permalink]

### Show Tags

25 Mar 2010, 12:55
11
6
D

1) N = w*1000 + x*100 + y*10 + z -->
N = (w + w*999) + (x + x*99) + (y + y*9) + z -->
N = (w + x + y + z) + 9*(w*111+x*11+y) --->
N = 13 + 9*(w*111+x*11+y) = 9 + 4 + 9*(w*111+x*11+y) --> remainder is 4. Sufficient

2) N + 5 = 9k --> N = 9k - 5 --> N = 9(k-1) + 4 --> remainder is 4. Sufficient
_________________
HOT! GMAT Club Forum 2020 | GMAT ToolKit 2 (iOS) - The OFFICIAL GMAT CLUB PREP APPs, must-have apps especially if you aim at 700+ | Limited Online GMAT/GRE Math tutoring
Manager
Joined: 28 Feb 2012
Posts: 103
GPA: 3.9
WE: Marketing (Other)
Re: If w, x, y, and z are the digits of the four-digit number N, a positiv  [#permalink]

### Show Tags

07 Jun 2013, 01:20
1
gmattokyo wrote:
If w, x, y, and z are the digits of the four-digit number N, a positive integer, what is the remainder when N is divided by 9?

(1) w + x + y + z = 13
(2) N + 5 is divisible by 9

There is a divisibility rule for 9 - if the sum of the digits of the number is divisible to 9 then the whole number is divisible to 9.

(1) the sum of the digits is 13 which is 4 more than 9, it means in order to be divisible to 9 one of the digits or the sum of some digits within the number should be 4 less. Otherwise there will be 4 extra when we divide to 9. So the remainder is 4. Sufficient.

(2) the same rule as in the first statement applies here as well. Moreover it does not really matter how big is the number whether four digits or two. For example lets take possible two digits numbers for N: 13, 22, 31 etc. in all case the remainder when divided by 9 will be 4. Or four digit numbers: 1129, or 1138, we will have the same remainder. Sufficient.
_________________
If you found my post useful and/or interesting - you are welcome to give kudos!
Manager
Joined: 17 Aug 2015
Posts: 96
Location: India
Concentration: Strategy, General Management
Schools: Duke '19 (II)
GMAT 1: 750 Q49 V42
GPA: 4
WE: Information Technology (Investment Banking)
Re: If w, x, y, and z are the digits of the four-digit number N, a positiv  [#permalink]

### Show Tags

05 Jan 2016, 23:54
1
timothyhenman1 wrote:
If w, x, y, and z are the digits of the four-digit number N, a positive integer, what is the remainder when N is divided by 9?

(1) w + x + y + z = 13
(2) N + 5 is divisible by 9

Nice question. Answer is (D) - Each alone is sufficient. Here's why:

(1) SUFFICIENT.

The rule for divisibility by 9 is that you add up all the digits of the given number. If this addition is divisible by nine, the number too is divisible by nine. In case the addition works out to be a number > 9, add the digits again. The same works for remainders - for 9, the remainder you get when you add digits and divide the result by 9 is the same remainder you get after dividing the original number by 9.

In our case, w+x+y+z = 13
Add digits again - 1+3 = 4
So, remainder after dividing by 9 is exactly 4 => SUFFICIENT.

Now for the analytical part - WHY?

A four digit number can be represented as 1000w + 100x + 10y + z.

Let's take the Remainder operation on "number" and call it Rem(number/9). Note that we can add remainders as long as we make the final adjustment to make them < original number (which is 9 here).

So, overall remainder is = Rem(1000w/9) + Rem (100x/9) + Rem(10y/9) +z

For any multiple of 10, you'll notice that 10^n - 1 is always divisible by 9 (e.g. 9, 99, 999 and so on). So, Rem(1000w/9) becomes Rem(1000/9)*Rem(w/9) = 1*Rem(w/9)
As w is a single digit, Rem(w/9) = w (even if w is 9, remainder can be technically 9. we will make the adjustment in the final phase)

Similarly, Rem(100x/9) = x, Rem (10y/9) = y and Rem(z/9) = z

So, overall remainder = Rem{(w+x+y+z)/9}
That's what we did above. We took the remainder after adding all the digits. This is complicated to remember, so just focus on the part before the incurably curious mind asks "WHY". It's a rule worth remembering.

(2) SUFFICIENT

If N+5 is divisible by 9, it means N+5 = 9k (where k is any integer). You can just stop here and say that this is sufficient, as if you know this then intuitively you know that N+4 would leave a remainder of 8, N+3 would leave 7 and so on.. with N/9 leaving a remainder of 4. Hence, SUFFICIENT.

Here's a more analytical treatment (and unnecessary at that - you MUST save time on DS if you can. If you are sure with the above approach, don't even do this) -

So, N = 9k - 5.

i.e. N/9 = k -5/9

compare this with Dividend / Divisor = Quotient + Remainder/Divisor

We get Remainder = -5.

For negative remainders, we add back the divisor to get the proper positive remainder. So, remainder = -5 +9 = 4. Therefore, SUFFICIENT.
_________________
If you like this post, be kind and help me with Kudos!

Cheers!
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8027
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: If w, x, y, and z are the digits of the four-digit number N, a positiv  [#permalink]

### Show Tags

06 Jan 2016, 18:36
1
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If w, x, y, and z are the digits of the four-digit number N, a positive integer, what is the remainder when N is divided by 9?

(1) w + x + y + z = 13
(2) N + 5 is divisible by 9

When you divide some integer n with 9, the remainder is same as the remainder which is the sum of all digits of the integer n and divided by 9. In the original condition, there is 1 variable(n), which should match with the number of equations. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.
In 1), the sum of all digits of n is 13. From 13=9*1+4, 4 is the remainder as divided by 9, which is unique and sufficient.
In 2), when dividing n+5 with 9, from n+5=9m, n=9m-5=9(m-1)+4, 4 is also the remainder as dividing n with 9, which is unique and sufficient. Therefore, the answer is D.

 For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only \$79 for 1 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Non-Human User
Joined: 09 Sep 2013
Posts: 13411
Re: If w, x, y, and z are the digits of the four-digit number N, a positiv  [#permalink]

### Show Tags

07 Aug 2019, 03:12
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: If w, x, y, and z are the digits of the four-digit number N, a positiv   [#permalink] 07 Aug 2019, 03:12
Display posts from previous: Sort by