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135 is a factor of 2^a.3^b.5^c

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by sanju09 » Thu Jul 31, 2014 3:51 am
If 135 is a factor of 2^a.3^b.5^c, where a, b, and c are non negative integers, which of the following CANNOT be the value of abc?
A. 8
B. 6
C. 4
D. 2
E. 0


Made Up!
Last edited by sanju09 on Sat Aug 02, 2014 10:16 pm, edited 2 times in total.
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by akhilsuhag » Thu Jul 31, 2014 4:36 am
c) 8
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by theCodeToGMAT » Thu Jul 31, 2014 7:47 am
135 = 5 x 3 x 3 x 3

2^a * 3^b * 5^c * _
Minimum value of
a = 0
b = 3
c = 1

To find: a*b*c

{A} 3 = 1 x 3 x 1 YES
{B} 6 = 1 x 3 x 2 YES
{C} 8 = 2 x 2 x 2 "we need at least 3 3's"
{D} 9 = 1 x 3 x 3 YES
{E} 12 = 2 x 3 x 2 YES

[spoiler]{C}[/spoiler]
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by GMATinsight » Thu Jul 31, 2014 8:34 am
sanju09 wrote:If 135 is a factor of 2^a.3^b.5^c, where a, b, and c are positive integers, which of the following CANNOT be the value of abc?
A. 3
B. 6
C. 8
D. 9
E. 12


Made Up!
135 is a factor of 2^a.3^b.5^c
135 = 3x3x3x5 = 3^3 x 5^1

therefore, 3^3 x 5^1 is a factor of 2^a.3^b.5^c
i.e. Comparing both sides
b must be 3 or a greater Integer and c must be 1 or a greater integer
therefore, abc must be greater than 3*1

Let's Check the options
A) 3
if a = 1, b = 3 and c = 1
2^a.3^b.5^c = 2 x 3^3 x 5^1 = 270 and 135 is factor if 270
So WRONG OPTION

B) 6
if a = 2, b = 3 and c = 1
2^a.3^b.5^c = 2^2 x 3^3 x 5^1 = 540 and 135 is factor if 540
So WRONG OPTION

C) 8
if a = 2, b = 4 and c = 1
2^a.3^b.5^c = 2^2 x 3^4 x 5^1 = 1620 and 135 is factor if 1620
So WRONG OPTION

D) 9
if a = 3, b = 3 and c = 1
2^a.3^b.5^c = 2^3 x 3^3 x 5^1 = 1080 and 135 is factor if 1080
So WRONG OPTION

E) 12
if a = 4, b = 3 and c = 1
2^a.3^b.5^c = 2^4 x 3^3 x 5^1 = 2160 and 135 is factor if 2160
So WRONG OPTION

All Options seem Wrong.... Ouch!!!
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by Matt@VeritasPrep » Thu Jul 31, 2014 10:16 am
(F):: none of the above.

For 135 to be a factor of 2ᵃ * 3ᵇ * 5ᶜ, we need b ≥ 3 and c ≥ 1. So ANY integer ≥ 3 will suffice: simply make c = 1 and b = the integer itself.
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by [email protected] » Thu Jul 31, 2014 3:28 pm
Hi sanju09,

This is a poorly designed question. I understand what you were "going for" (prime factorization rules and how you can use them to figure out possibilities), but the question isn't properly designed. Here's why:

The prime factors of 135 are 3x3x3x5.

We're told that A, B and C are positive integers. Since 135 is a factor of (2^A)(3^B)(5^C), we know that....

A can be ANY positive integer
B can be any positive integer greater than or equal to 3
C can be ANY positive integer

If we choose, A=1, C=1, then B can be ANY integer greater than or equal to 3....

Thus, ABC can equal 3, 4, 5, 6, 7, 8 etc. So all of the answers are possible.

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by Matt@VeritasPrep » Sat Aug 02, 2014 11:14 am
Rich, there's no need to pile on Sanju here. I made the exact same point in my post - give the guy a break!
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by sanju09 » Sat Aug 02, 2014 10:08 pm
OOPS...

Bad selection of answer choices. Edited. Thanks for the feed backs. :lol:
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by sanju09 » Sat Aug 02, 2014 10:11 pm
Matt@VeritasPrep wrote:Rich, there's no need to pile on Sanju here. I made the exact same point in my post - give the guy a break!
Nice prescription Matt :wink:

Question needs corrections at two places, one in wordings, "a, b, and c are non negative integers", and the other, of course in the answer choices. Thanks for waking me up.

Please re-read it as below:

If 135 is a factor of 2^a.3^b.5^c, where a, b, and c are non negative integers, which of the following CANNOT be the value of abc?
A. 8
B. 6
C. 4
D. 2
E. 0
The mind is everything. What you think you become. -Lord Buddha



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by sanju09 » Sat Aug 02, 2014 10:25 pm
GMATinsight wrote:
sanju09 wrote:If 135 is a factor of 2^a.3^b.5^c, where a, b, and c are positive integers, which of the following CANNOT be the value of abc?
A. 3
B. 6
C. 8
D. 9
E. 12


Made Up!
135 is a factor of 2^a.3^b.5^c
135 = 3x3x3x5 = 3^3 x 5^1

therefore, 3^3 x 5^1 is a factor of 2^a.3^b.5^c
i.e. Comparing both sides
b must be 3 or a greater Integer and c must be 1 or a greater integer
therefore, abc must be greater than 3*1

Let's Check the options
A) 3
if a = 1, b = 3 and c = 1
2^a.3^b.5^c = 2 x 3^3 x 5^1 = 270 and 135 is factor if 270
So WRONG OPTION

B) 6
if a = 2, b = 3 and c = 1
2^a.3^b.5^c = 2^2 x 3^3 x 5^1 = 540 and 135 is factor if 540
So WRONG OPTION

C) 8
if a = 2, b = 4 and c = 1
2^a.3^b.5^c = 2^2 x 3^4 x 5^1 = 1620 and 135 is factor if 1620
So WRONG OPTION

D) 9
if a = 3, b = 3 and c = 1
2^a.3^b.5^c = 2^3 x 3^3 x 5^1 = 1080 and 135 is factor if 1080
So WRONG OPTION

E) 12
if a = 4, b = 3 and c = 1
2^a.3^b.5^c = 2^4 x 3^3 x 5^1 = 2160 and 135 is factor if 2160
So WRONG OPTION

All Options seem Wrong.... Ouch!!!
Very genuine Ouch!!! Sir, thanks and congratulations for spotting it before others.
The mind is everything. What you think you become. -Lord Buddha



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