Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack

 It is currently 29 May 2017, 00:51

### 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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# If a is a positive integer and 20 < a < 100, is a^3 − a

Author Message
TAGS:

### Hide Tags

Moderator
Joined: 01 Sep 2010
Posts: 3182
Followers: 861

Kudos [?]: 7331 [1] , given: 1065

If a is a positive integer and 20 < a < 100, is a^3 − a [#permalink]

### Show Tags

29 Aug 2013, 08:54
1
KUDOS
5
This post was
BOOKMARKED
00:00

Difficulty:

55% (hard)

Question Stats:

67% (02:27) correct 33% (01:55) wrong based on 143 sessions

### HideShow timer Statistics

If a is a positive integer and 20 < a < 100, is a^3 − a divisible by 15?

(1) The units digit of a is 9.
(2) The tens digit of a is 3.
[Reveal] Spoiler: OA

_________________

Last edited by Bunuel on 29 Aug 2013, 08:56, edited 1 time in total.
Math Expert
Joined: 02 Sep 2009
Posts: 39042
Followers: 7751

Kudos [?]: 106500 [1] , given: 11626

Re: If a is a positive integer and 20 < a < 100, is a 3 − a [#permalink]

### Show Tags

29 Aug 2013, 09:04
1
KUDOS
Expert's post
If a is a positive integer and 20 < a < 100, is a^3 − a divisible by 15?

In order an integer to be divisible by 15 it should be divisible by both 3 and 5.

(1) The units digit of a is 9 --> $$a^3-a=(a-1)a(a+1)$$ --> out of 3 consecutive integers ((a-1), a, and (a+1)), one must be divisible by 3 and as the units digit of a is 9, then the units digit of a+1 is 0, so a+1 is divisible by 5. Thus a^3-a is divisible by 15. Sufficient.

(2) The tens digit of a is 3. If a=30, then the answer is YES but if a=32 ((a-1)a(a+1)=31*32*33), then the answer is NO. Not sufficient.

_________________
Current Student
Joined: 21 Oct 2013
Posts: 193
Location: Germany
GMAT 1: 660 Q45 V36
GPA: 3.51
Followers: 1

Kudos [?]: 40 [0], given: 19

Re: If a is a positive integer and 20 < a < 100, is a^3 − a [#permalink]

### Show Tags

04 Mar 2014, 04:08
(1) units digit of a is 9. If you raise 9 to some power, it repeats this pattern: 9^1=9 9^2= 81 9^3 =...9. Hence, if we raise a number with the units digit 9 to the power of 3, the outcome will also have the units digit of 9. If we then substract the number with the units digit 9, the outcome will have the units digit 0. Hence div. by 15, SUFF.

(2) Tens digit a three: It's thus 31,32,33...or 39. 31^3-31 would be div. by 15, 32^3 - 32 not. IS.

Senior Manager
Joined: 20 Dec 2013
Posts: 267
Location: India
Followers: 0

Kudos [?]: 89 [0], given: 29

Re: If a is a positive integer and 20 < a < 100, is a^3 − a [#permalink]

### Show Tags

04 Mar 2014, 05:26
From S1:We can infer that a^3-a=(a-1)*a*(a+1) which are 3 consecutive integers.So they're always divisible by 3 in the range of 'a' given.We just need to prove that any one or more of these will have 5 as a factor too.
S1 gives that 'a' ends with 9 so (a+1) will end with 0.always div by 5.Hence suff.

From S2:we get that 'a' will be in the range of 30s.We could consider 2 simple egs. (30,31,32)-Ans Yes.
While (31,32,33)-And No.Not sufficient.

Option A.

Posted from my mobile device
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15504
Followers: 651

Kudos [?]: 210 [0], given: 0

Re: If a is a positive integer and 20 < a < 100, is a^3 − a [#permalink]

### Show Tags

19 Mar 2015, 12:14
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.
_________________
BSchool Forum Moderator
Joined: 12 Aug 2015
Posts: 2135
Followers: 74

Kudos [?]: 615 [0], given: 558

Re: If a is a positive integer and 20 < a < 100, is a^3 − a [#permalink]

### Show Tags

15 Mar 2016, 10:21
Here If we write doen A^3-A as A (A+1)(A-1) => we can say that A is sufficient
remember we need to check only 3 values for the pattern of 3
hence 39=> yes
49=> yes
59=> yes
hence A
statement 2 can be discarded as for 39 => yes
else for 32 => NO
_________________

Give me a hell yeah ...!!!!!

Re: If a is a positive integer and 20 < a < 100, is a^3 − a   [#permalink] 15 Mar 2016, 10:21
Similar topics Replies Last post
Similar
Topics:
2 Is integer x<−20? 1 06 Feb 2016, 22:17
5 If n is an integer and 100 < n <200, what is the value of n? 9 05 Oct 2016, 23:03
1 The positive integers are such that p < q r < s < 3 20 May 2011, 06:32
6 If n is an integer and 100 < n < 200, what is the value 11 23 Feb 2017, 07:02
If n is an integer and 100 < n <200, what is the value 2 01 Jan 2010, 17:37
Display posts from previous: Sort by