GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 03 Jun 2020, 21:29 ### 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

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If m is a positive integer, then m^3 has how many digits?

Author Message
TAGS:

### Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 64234
If m is a positive integer, then m^3 has how many digits?  [#permalink]

### Show Tags

2
58 00:00

Difficulty:   95% (hard)

Question Stats: 52% (02:12) correct 48% (02:07) wrong based on 1281 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 Bunuel on 21 Dec 2010, 13:46.
Last edited by Bunuel on 26 Apr 2020, 09:10, edited 3 times in total.
Edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 64234
Re: If m is a positive integer, then m^3 has how many digits?  [#permalink]

### Show Tags

13
38
SOLUTION

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: 77
Location: Bangalore, India
Re: If m is a positive integer, then m^3 has how many digits?  [#permalink]

### Show Tags

17
5
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, 04:06.
Last edited by muralimba on 22 Dec 2010, 06:14, edited 1 time in total.
##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 64234
Re: If m is a positive integer, then m^3 has how many digits?  [#permalink]

### Show Tags

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 V
Joined: 16 Oct 2010
Posts: 10497
Location: Pune, India
Re: If m is a positive integer, then m^3 has how many digits?  [#permalink]

### Show Tags

11
5
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

Intern  Joined: 10 Apr 2012
Posts: 37
Concentration: Finance
Schools: Goizueta '19 (I)
WE: Analyst (Commercial Banking)
Re: If m is a positive integer, then m^3 has how many digits?  [#permalink]

### Show Tags

2
1
Number Picking Strategy:

1)m can have three digits but m^3 differ e.g 100 and 999.Insufficient
2) m^2 can have 5 digits but m^3 differ e.g 100 and 315. Insufficient
Together, the statements are insufficient. Take 100 and 315 as in the (2) above.

Intern  Joined: 20 May 2014
Posts: 34
Location: India
Schools: IIMC
GMAT 1: 700 Q51 V32
Re: If m is a positive integer, then m^3 has how many digits?  [#permalink]

### Show Tags

3
Hi MensaNumber,

I hope this helps,

Statement 1:
M has 3 digits

M can be any number between 100 and 999,
Number of digits in $$100^3$$ = 7
Number of digits in $$1000^3$$ = 10 => Hence $$999^3$$ will have 9 digits

So, number of digits could be 7, 8 or 9

Statement 1 not sufficient

Statement 2:
$$M^2$$ has 5 digits

Smallest M is 100, where$$M^2$$ has 5 digits
Largest M would be close to 300, as $$300^2$$ = 90000

M=100, will give number of digits in $$M^3$$ as 7
M = 300, will give number of digits in $$M^3$$ as 8

So, number of digits could be 7 or 8

Statement 2 not sufficient

Combining both statement,
Number of digits could be 7 or 8

Therefore,
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 16775
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

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
_________________
DS Forum Moderator V
Joined: 19 Oct 2018
Posts: 1880
Location: India
Re: If m is a positive integer, then m^3 has how many digits?  [#permalink]

### Show Tags

Statement 1-
2≤log m<3
6≤ 3log m<9

Number of digits in $$m^3$$ can be 7, 8 or 9
Insufficient

Statement 2-

4≤ 2log m< 5
6≤ 3log m< 7.5

Number of digits in $$m^3$$ can be 7 or 8
Insufficient

Combing both equations
Number of digits in $$m^3$$ can be 7 or 8

Insufficient

E

Bunuel 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.

Data Sufficiency
Question: 104
Category: Arithmetic Properties of numbers
Page: 160
Difficulty: 650

The Official Guide For GMAT® Quantitative Review, 2ND Edition
Non-Human User Joined: 09 Sep 2013
Posts: 15060
Re: If M is a positive integer, then M^3 has how many digits?  [#permalink]

### Show Tags

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 M is a positive integer, then M^3 has how many digits?   [#permalink] 26 Apr 2020, 09:08

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