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

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Math Expert
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If m is a positive integer, then m^3 has how many digits? [#permalink]

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20 Feb 2014, 00:17
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The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

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

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20 Feb 2014, 00:17
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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.

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

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20 Feb 2014, 00:31
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.

Since the constraints are true for a range of values of m (between 100 and $$[\sqrt{100000}]$$, inclusive), this must be true for any m satisfying (1) and (2). Note that m=100 satisfies both. Since 100^3 has seven digits, m^3 must have seven digits.
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Re: If m is a positive integer, then m^3 has how many digits? [#permalink]

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20 Feb 2014, 23:20
Option E.
From S1:Not suff since if m=100 ; m^3=7 digits
And if m=900;m^3=9 digits.

From S2:m^2 has 5 digits=> 99<m<316
m=100;m^3=7 digits.
m=315;m^3=8 digits.

Combining both the statements.
m=3 digits
and m^2=5 digits.
Same situation as in case 2.Not suff.
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Re: If m is a positive integer, then m^3 has how many digits? [#permalink]

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21 Feb 2014, 05:39
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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.

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

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21 May 2014, 08:26
Hi Bunuel, Is this an odd ball? Or are there similar problems available? It took me 3:22 mins to solve it. I definitely need more practice. Thanks!
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Re: If m is a positive integer, then m^3 has how many digits? [#permalink]

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22 May 2014, 01:49
MensaNumber wrote:
Hi Bunuel, Is this an odd ball? Or are there similar problems available? It took me 3:22 mins to solve it. I definitely need more practice. Thanks!

I wasn't able to find any similar questions but below are all questions we have on digits:
All DS Fractions/Ratios/Decimals questions: search.php?search_id=tag&tag_id=36
All PS Fractions/Ratios/Decimals questions: search.php?search_id=tag&tag_id=57

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

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22 May 2014, 10:51
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Oh Ok. Thanks for checking tho'

Will take a look at the links you sent. Thanks again
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Re: If m is a positive integer, then m^3 has how many digits? [#permalink]

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23 May 2014, 03:02
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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,
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Re: If m is a positive integer, then m^3 has how many digits? [#permalink]

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27 Nov 2017, 12:03
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

[Reveal] Spoiler:
E

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Re: If m is a positive integer, then m^3 has how many digits?   [#permalink] 27 Nov 2017, 12:03
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