Problem Solving

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!

c) 8

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]

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!!!

(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.

Rich, there's no need to pile on Sanju here. I made the exact same point in my post - give the guy a break!

OOPS...

Bad selection of answer choices. Edited. Thanks for the feed backs.

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

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

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.