NEW HERE? LEARN MORE
closeclose
Last visit was: 24 Apr 2024, 22:24 It is currently 24 Apr 2024, 22:24
Decision Tracker
My Rewards
New posts
New comers' posts

Close
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
Your Progress

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
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

If positive integer p divided by 9 leaves a remainder of 1, which of

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 92900
Own Kudos [?]: 618847 [7]
Given Kudos: 81588
Send PM
CAT Tests Forum Quiz Mode
If positive integer p divided by 9 leaves a remainder of 1, which of [#permalink] New post   Updated on: 05 Jul 2019, 03:28
1
Kudos
6
Bookmarks
Expert Reply
00:00
A
B
C
D
E

Difficulty:

15% (low)

Question Stats:

76% (01:00) correct 24% (01:26) wrong based on 123 sessions
Hide Show timer Statistics
If positive integer p divided by 9 leaves a remainder of 1, which of the following must be true?

I. p is even.
II. p is odd.
III. p = 3z + 1 for some integer z.

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

Source: Nova GMAT
Difficulty Level: 500
ShowHide Answer
Official Answer
C

Originally posted by Bunuel on 15 Jun 2017, 05:15.
Last edited by Sajjad1994 on 05 Jul 2019, 03:28, edited 1 time in total.
Added Source
User avatar
L
sashiim20
Director
Director
Joined: 04 Dec 2015
Posts: 620
Own Kudos [?]: 1585 [1]
Given Kudos: 276
Location: India
Concentration: Technology, Strategy
Schools: ISB '19 IIMB IIMA XLRI HEC Sept19 intake Rotman '21 NUS '21 NTU '20 Trinity MBA '20 Bocconi '21 Smurfit "21
WE:Information Technology (Consulting)
Send PM
GMAT ToolKit User
Re: If positive integer p divided by 9 leaves a remainder of 1, which of [#permalink] New post   15 Jun 2017, 05:35
1
Bookmarks
Bunuel wrote:
If positive integer p divided by 9 leaves a remainder of 1, which of the following must be true?

I. p is even.
II. p is odd.
III. p = 3z + 1 for some integer z.

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only


The number is in the form, \(p = 9x + 1\) --------- (x is the quotient )
Hence number when divided by 3 will leave remainder 1.
Therefore number must be in the form; \(p = 3z + 1\) ---------- (z is the quotient) (III)

Lets try even and odd numbers for p.
If p is 10, divided by 9 will leave remainder 1.
If p is 19 , divided by 9 will leave remainder 1.

Therefore p could be even or odd. Therefore I and II must not be true.

Only III. Answer C...
User avatar
L
Abhishek009
Board of Directors
Joined: 11 Jun 2011
Status:QA & VA Forum Moderator
Posts: 6072
Own Kudos [?]: 4689 [0]
Given Kudos: 463
Location: India
GPA: 3.5
WE:Business Development (Commercial Banking)
Send PM
GMAT ToolKit User
Re: If positive integer p divided by 9 leaves a remainder of 1, which of [#permalink] New post   15 Jun 2017, 08:03
Bunuel wrote:
If positive integer p divided by 9 leaves a remainder of 1, which of the following must be true?

I. p is even.
II. p is odd.
III. p = 3z + 1 for some integer z.

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only


Values of p can be 1 , 10 , 19 , 28.........

I. p is even (Can be both even/odd)
II. p is odd (Can be both even/odd)
III. p = 3z + 1 for some integer z - True in all possible cases...

Thus, answer will (C)
User avatar
L
generis
Senior SC Moderator
Joined: 22 May 2016
Posts: 5330
Own Kudos [?]: 35486 [0]
Given Kudos: 9464
Send PM
CAT Tests
Re: If positive integer p divided by 9 leaves a remainder of 1, which of [#permalink] New post   15 Jun 2017, 08:34
Expert Reply
Bunuel wrote:
If positive integer p divided by 9 leaves a remainder of 1, which of the following must be true?

I. p is even.
II. p is odd.
III. p = 3z + 1 for some integer z.

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

Try to disprove each statement.

\(\frac{p}{9}\) = a + 1, so p = 9a + 1

p = 10, 19, 28, 37, 46, 55 ... (leaving out 1 on purpose)

From list, neither I nor II is true. 19, odd, and 10, even, disprove each respectively. At this point you could choose (C) and move on, because all other answers contain I and/or II

III. p = 3z + 1 for some integer z

10 = 3(3) + 1.
19 = 3(6) + 1.
55 = 3(18) + 1.
III is true. Answer C
User avatar
L
ScottTargetTestPrep
Target Test Prep Representative
Joined: 14 Oct 2015
Status:Founder & CEO
Affiliations: Target Test Prep
Posts: 18756
Own Kudos [?]: 22049 [2]
Given Kudos: 283
Location: United States (CA)
Send PM
Re: If positive integer p divided by 9 leaves a remainder of 1, which of [#permalink] New post   17 Jun 2017, 05:09
2
Bookmarks
Expert Reply
Bunuel wrote:
If positive integer p divided by 9 leaves a remainder of 1, which of the following must be true?

I. p is even.
II. p is odd.
III. p = 3z + 1 for some integer z.

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only



Since positive integer p divided by 9 leaves a remainder of 1, we have:

p/9 = Q + 1/9

p = 9Q + 1

Thus, p can be values such as 1, 10, 19, 28, ….

Thus, p can be either even or odd.

Now we can look at Roman numeral III:

p = 3z + 1 for some integer z.

Recall that we have expressed that p = 9Q + 1, so p = 3(3Q) + 1. We see that if we let z = 3Q, we can express p as 3z + 1. So Roman numeral III is true.

Answer: C
avatar
S
sonikavadhera
Manager
Manager
Joined: 06 Nov 2016
Posts: 94
Own Kudos [?]: 48 [0]
Given Kudos: 15
Location: India
Schools: Tepper '20 (A$) Rotman '20 (A$) Marshall '20 (A) ISB '19 (A)
GMAT 1: 710 Q50 V36
GPA: 2.8
Send PM
Reviews Badge
Re: If positive integer p divided by 9 leaves a remainder of 1, which of [#permalink] New post   21 Jun 2017, 02:07
P CAN BE 1,10,19....

hence I and II can be true or false

III is true for all cases

Option C
avatar
B
FB2017
Manager
Manager
Joined: 18 Jun 2017
Posts: 50
Own Kudos [?]: 13 [0]
Given Kudos: 165
Send PM
Re: If positive integer p divided by 9 leaves a remainder of 1, which of [#permalink] New post   16 Aug 2017, 08:57
p=3z+1 should be true where 1 is the remainder as per problem statement.
Since it is divisible by 9 z is quotient and the above expression cabe written as multiple of 3.
Option C.
User avatar
bumpbot
Non-Human User
Joined: 09 Sep 2013
Posts: 32657
Own Kudos [?]: 821 [0]
Given Kudos: 0
Send PM
Re: If positive integer p divided by 9 leaves a remainder of 1, which of [#permalink] New post   22 Mar 2024, 13:23
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.
GMAT Club Bot
gmatclubot
Re: If positive integer p divided by 9 leaves a remainder of 1, which of [#permalink]
22 Mar 2024, 13:23
Moderators:
Bunuel
Math Expert
92900 posts
yashikaaggarwal
Senior Moderator - Masters Forum
3137 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions