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

 It is currently 15 Oct 2019, 00:19

### 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 x, y, and z are integers such that 67500 is divisible by (2^x)(3^

Author Message
TAGS:

### Hide Tags

GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
If x, y, and z are integers such that 67500 is divisible by (2^x)(3^  [#permalink]

### Show Tags

26 Feb 2019, 15:03
00:00

Difficulty:

75% (hard)

Question Stats:

52% (03:20) correct 48% (03:11) wrong based on 33 sessions

### HideShow timer Statistics

GMATH practice exercise (Quant Class 16)

If x, y, and z are integers such that 67500 is divisible by (2^x)(3^y)(5^z) and (2^x)(3^y)(5^z) is NOT a multiple of 54, what is the maximum possible value of 3x+2y+z?

(A) 15
(B) 14
(C) 13
(D) 12
(E) less than 12

_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Current Student
Affiliations: All Day Test Prep
Joined: 08 May 2018
Posts: 75
Location: United States (IL)
Schools: Booth '20 (A)
GMAT 1: 770 Q51 V49
GRE 1: Q167 V167
GPA: 3.58
Re: If x, y, and z are integers such that 67500 is divisible by (2^x)(3^  [#permalink]

### Show Tags

26 Feb 2019, 15:59
2
fskilnik wrote:
GMATH practice exercise (Quant Class 16)

If x, y, and z are integers such that 67500 is divisible by (2^x)(3^y)(5^z) and (2^x)(3^y)(5^z) is NOT a multiple of 54, what is the maximum possible value of 3x+2y+z?

(A) 15
(B) 14
(C) 13
(D) 12
(E) less than 12

Step 1: Break up 67500 into it's prime factors and find out how many 2's, 3's, and 5's we have.
- 67500 = (2^2)(3^3)(5^4)

Step 2: Make sure our factors don't include a 54
- 54 = (3^3)(2)
- We want to use take out the MINIMUM value of prime factors possible(leave in the max possible factor) so that they no longer contain a 54
- This means we can take out two of our 2's or one of our 3's.
- Since 3 < 2^2 take out a 3.

Step 3: Max Factor = Whatever Prime Factors are left
- (2^2)(3^2)(5^4)
- therefore x =2, y =2, z =4

Step 4: Calculate 3x + 2y + z
- 6 + 4 + 4 = 14

_________________
University of Chicago Booth School of Business, Class of 2020

Unlimited private GMAT Tutoring in Chicago for less than the cost a generic prep course. No tracking hours. No watching the clock.

GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
Re: If x, y, and z are integers such that 67500 is divisible by (2^x)(3^  [#permalink]

### Show Tags

27 Feb 2019, 08:08
fskilnik wrote:
GMATH practice exercise (Quant Class 16)

If x, y, and z are integers such that 67500 is divisible by (2^x)(3^y)(5^z) and (2^x)(3^y)(5^z) is NOT a multiple of 54, what is the maximum possible value of 3x+2y+z?

(A) 15
(B) 14
(C) 13
(D) 12
(E) less than 12

Very nice, arosman. Thank you for your contribution (and kudos)!

Our official solution follows:

$$? = \max \left( {3x + 2y + z} \right)\,\,\,\left( * \right)$$

$$x,y,z\,\,\mathop \ge \limits^{\left( * \right)} \,\,0\,\,\,{\rm{ints}}\,\,\,\left( {**} \right)$$

$$67500 = \underleftrightarrow {675 \cdot 100} = 25 \cdot 27 \cdot 4 \cdot 25 = {2^2} \cdot {3^3} \cdot {5^4}$$

$$54 = 2 \cdot 27 = 2 \cdot {3^3}$$

$$\left. \matrix{ {\mathop{\rm int}} = {{\,{2^2} \cdot {3^3} \cdot {5^4}\,} \over {{2^x} \cdot {3^y} \cdot {5^z}}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\left\{ \matrix{ \,0 \le x \le 2 \hfill \cr \,0 \le y \le 3 \hfill \cr \,0 \le z \le 4 \hfill \cr} \right. \hfill \cr {\mathop{\rm int}} \ne {{\,{2^x} \cdot {3^y} \cdot {5^z}\,} \over {2 \cdot {3^3}}}\,\,\,\, \Rightarrow \,\,\,\,x = 0\,\,{\rm{or}}\,\,y < 3\,\,\left( {{\rm{or}}\,\,{\rm{both}}} \right)\,\, \hfill \cr} \right\}\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{\rm{Take}}\,\,\,\left( {x,y,z} \right) = \left( {2,3 - 1,4} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 14$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Re: If x, y, and z are integers such that 67500 is divisible by (2^x)(3^   [#permalink] 27 Feb 2019, 08:08
Display posts from previous: Sort by