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

 It is currently 23 Sep 2018, 09:29

### 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 m is a positive integer, then m^3 has how many digits?

Author Message
TAGS:

### Hide Tags

Intern
Joined: 04 Jun 2010
Posts: 5
If m is a positive integer, then m^3 has how many digits?  [#permalink]

### Show Tags

Updated on: 12 Oct 2013, 05:26
12
00:00

Difficulty:

85% (hard)

Question Stats:

52% (01:30) correct 48% (01:27) wrong based on 368 sessions

### HideShow timer Statistics

If m is a positive integer, then m^3 has how many digits?

(1) m has 3 digits
(2) m^2 has 5 digits

Originally posted by Runirish on 21 Dec 2010, 14:46.
Last edited by Bunuel on 12 Oct 2013, 05:26, edited 1 time in total.
Edited the question.
Math Expert
Joined: 02 Sep 2009
Posts: 49320

### Show Tags

21 Dec 2010, 15:15
9
8
Runirish wrote:
If m is a positive integer, then m^3 has how many digits?
1. m has 3 digits
2. m^2 has 5 digits

How would you do this quickly? Is there a rule that I am unaware of? I could do it, but I had to pick a few numbers. Thanks.

If m is a positive integer, then m^3 has how many digits?

Pick some easy numbers.

(1) m has 3 digits --> if $$m=100=10^2$$ then $$m^3=10^6$$ so it will have 7 digits but if $$m=300=3*10^2$$ then $$m^3=27*10^6$$ so it will have 8 digits. Not sufficient.

(2) m^2 has 5 digits --> the same values of $$m$$ (100 and 300) satisfy this statement too (because if $$m=10^2$$ then $$m^2=10^4$$ and has 5 digits and if $$m=3*10^2$$ then $$m^2=9*10^4$$ also has 5 digits), so $$m^3$$ may still have 7 or 8 digits. Not sufficient.

(1)+(2) The same example worked for both statements so even taken together statements are not sufficient.

_________________
Manager
Joined: 30 Aug 2010
Posts: 88
Location: Bangalore, India

### Show Tags

Updated on: 22 Dec 2010, 07:14
9
1
Runirish wrote:
If m is a positive integer, then m^3 has how many digits?
1. m has 3 digits
2. m^2 has 5 digits

How would you do this quickly? Is there a rule that I am unaware of? I could do it, but I had to pick a few numbers. Thanks.

as we know,

the minimum value of a 3 digit integer is 100 = $$10^2$$
the maximum value of a 3 digit integer is 999 = $$10^4 - 1$$
the minimum value of a 5 digit integer is 10000 = $$10^4$$
the maximum value of a 5 digit integer is 99999 = $$10^6 - 1$$
.
.
hence,
.

the minimum value of a $$n$$ digit integer is $$10^(n-1)$$
the maximum value of a $$n$$ digit integer is $$10^(n+1) - 1$$

Back to original qtn:

If m is a positive integer, then $$m^3$$ has how many digits?
stmnt1: $$m$$ has 3 digits
==> $$10^2 <= m < 10^4$$
==> $$10^6 <= m^3 < 10^12$$
==> $$m^3$$ can have minimum of 7 (i.e 6+1) and max of 11 digits (i.e. 12-1)
hence NOT suff.

stmnt2: m^2 has 5 digits
==> $$10^4 <= m^2 < 10^6$$
==> $$10^2 <= m < 10^3$$
==> $$10^6 <= m^3 < 10^9$$
==> $$m^3$$ can have minimum of 7 (i.e 6+1) and max of 8 digits (i.e. 9-1)
hence NOT suff.

stmnt1&2 together: We can conclude that $$m^3$$ can have minimum of 6 and max of 8 digits(i.e. 12-1) ==> m can have 7 or 8 digits
hence NOT suff.

Regards,
Murali.
Kudos?

Originally posted by muralimba on 22 Dec 2010, 05:06.
Last edited by muralimba on 22 Dec 2010, 07:14, edited 1 time in total.
##### General Discussion
Math Expert
Joined: 02 Sep 2009
Posts: 49320

### Show Tags

22 Dec 2010, 06:21
1
muralimba wrote:
Runirish wrote:
If m is a positive integer, then m^3 has how many digits?
1. m has 3 digits
2. m^2 has 5 digits

How would you do this quickly? Is there a rule that I am unaware of? I could do it, but I had to pick a few numbers. Thanks.

as we know,

the minimum value of a 3 digit integer is 100 = $$10^2$$
the maximum value of a 3 digit integer is 999 = $$10^4 - 1$$
the minimum value of a 5 digit integer is 10000 = $$10^4$$
the maximum value of a 5 digit integer is 99999 = $$10^6 - 1$$
.
.
hence,
.

the minimum value of a $$n$$ digit integer is $$10^n$$
the maximum value of a $$n$$ digit integer is $$10^(n+1) - 1$$

Back to original qtn:

If m is a positive integer, then $$m^3$$ has how many digits?
stmnt1: $$m$$ has 3 digits
==> $$10^2 <= m < 10^4$$
==> $$10^6 <= m^3 < 10^12$$
==> $$m^3$$ can have minimum of 6 and max of 11 digits (i.e. 12-1)
hence NOT suff.

stmnt2: m^2 has 5 digits
==> $$10^4 <= m^2 < 10^6$$
==> $$10^2 <= m < 10^3$$
==> $$10^6 <= m^3 < 10^9$$
==> $$m^3$$ can have minimum of 6 and max of 8 digits (i.e. 9-1)
hence NOT suff.

stmnt1&2 together: We can conclude that $$m^3$$ can have minimum of 6 and max of 8 digits(i.e. 12-1) ==> m can have 6,7, or 8 digits
hence NOT suff.

Regards,
Murali.
Kudos?

m^3 can have only 7 or 8 digits, not 6. If m=100=10^2 then m^3=10^6 and it has 6 trailing zeros but 7 digits.
_________________
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8288
Location: Pune, India

### Show Tags

22 Dec 2010, 12:18
7
2
Runirish wrote:
If m is a positive integer, then m^3 has how many digits?
1. m has 3 digits
2. m^2 has 5 digits

How would you do this quickly? Is there a rule that I am unaware of? I could do it, but I had to pick a few numbers. Thanks.

1. m has 3 digits

When I look at such statements, I invariably think of the extremities. (as muralimba did above)
Smallest m = 100 which implies m^3 = 10^6 giving 7 digits.
Largest m = 999 but it is not easy to find its cube so I take a number close to it i.e. 1000 and find its cube which is 10^9 i.e. smallest 10 digit number. Hence 999^3 will have 9 digits.
Since we can have 7, 8 or 9 digits, this statement is not sufficient.

2. m^2 has 5 digits
Now try to forget what you read above. Just focus on this statement.
Smallest m^2 = 10000 which implies m = 100
Largest m^2 is less than 99999 which gives m as something above 300 but less than 400.
Now, if m is 100, m^3 = 10^6 giving 7 digits.
If m is 300, m^3 = 27000000 giving 8 digits.
Since we have 7 or 8 digits for m, this statement is not sufficient.

Now combining both, remember one important point - If one statement is already included in the other, and the more informative statement is not sufficient alone, both statements will definitely not be sufficient together.

e.g. statement 1 tells us that m has 3 digits. Statement 2 tells us that m is between 100 and 300 something, so statement 2 tells us that m has 3 digits (what statement 1 told us) and something extra (that its value lies between 100 and 300 something). Statement 2 is more informative and is not sufficient alone. Since statement 1 doesn't add any new information to statement 2, no way will they both together be sufficient.
_________________

Karishma
Veritas Prep GMAT Instructor

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Retired Moderator
Joined: 29 Oct 2013
Posts: 272
Concentration: Finance
GPA: 3.7
WE: Corporate Finance (Retail Banking)

### Show Tags

07 Apr 2014, 07:09
muralimba wrote:
Runirish wrote:
If m is a positive integer, then m^3 has how many digits?
1. m has 3 digits
2. m^2 has 5 digits

How would you do this quickly? Is there a rule that I am unaware of? I could do it, but I had to pick a few numbers. Thanks.

as we know,

the minimum value of a 3 digit integer is 100 = $$10^2$$
the maximum value of a 3 digit integer is 999 = $$10^4 - 1$$
the minimum value of a 5 digit integer is 10000 = $$10^4$$
the maximum value of a 5 digit integer is 99999 = $$10^6 - 1$$
.
.
hence,
.

the minimum value of a $$n$$ digit integer is $$10^(n-1)$$
the maximum value of a $$n$$ digit integer is $$10^(n+1) - 1$$

Back to original qtn:

If m is a positive integer, then $$m^3$$ has how many digits?
stmnt1: $$m$$ has 3 digits
==> $$10^2 <= m < 10^4$$
==> $$10^6 <= m^3 < 10^12$$
==> $$m^3$$ can have minimum of 7 (i.e 6+1) and max of 11 digits (i.e. 12-1)
hence NOT suff.

stmnt2: m^2 has 5 digits
==> $$10^4 <= m^2 < 10^6$$
==> $$10^2 <= m < 10^3$$
==> $$10^6 <= m^3 < 10^9$$
==> $$m^3$$ can have minimum of 7 (i.e 6+1) and max of 8 digits (i.e. 9-1)
hence NOT suff.

stmnt1&2 together: We can conclude that $$m^3$$ can have minimum of 6 and max of 8 digits(i.e. 12-1) ==> m can have 7 or 8 digits
hence NOT suff.

Regards,
Murali.
Kudos?

Murali, While you approach is conceptually solid, it is marred by silly errors. For instance, here "==> $$10^2 <= m < 10^4$$" it should be 10^2 <= m < 10^3 and hence m^3 can have no. of digits from 5-9. Another one was already pointed by Bunuel.
Thanks for sharing nonetheless.
_________________

My journey V46 and 750 -> http://gmatclub.com/forum/my-journey-to-46-on-verbal-750overall-171722.html#p1367876

EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 12432
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: If m is a positive integer, then m^3 has how many digits?  [#permalink]

### Show Tags

09 Dec 2017, 17:15
1
Hi All,

We're told that M is a positive integer. We're asked for the number of digits that M^3 has. This question can be solved by TESTing VALUES.

1) M has 3 digits.

IF....
M=100, then M^3 = 1,000,000 and the answer to the question is 7
M=300, then M^3 = 27,000,000 and the answer to the question is 8
Fact 1 is INSUFFICIENT

2) M^2 has 5 digits

The same values that we used in Fact 1 also 'fit' Fact 2:
M=100, M^2=10,000 and M^3 = 1,000,000 and the answer to the question is 7
M=300, M^2=90,000 and M^3 = 27,000,000 and the answer to the question is 8
Fact 2 is INSUFFICIENT

Combined, there's no more work needed. We already have two values that 'fit' both Facts and produce two different answers.
Combined, INSUFFICIENT

GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

# Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save \$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

Re: If m is a positive integer, then m^3 has how many digits? &nbs [#permalink] 09 Dec 2017, 17:15
Display posts from previous: Sort by

# If m is a positive integer, then m^3 has how many digits?

## Events & Promotions

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.