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# If x and y are integers and 72^x∗54^y=96, what is the value of x−y?

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Math Expert
Joined: 02 Sep 2009
Posts: 58452
If x and y are integers and 72^x∗54^y=96, what is the value of x−y?  [#permalink]

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07 Mar 2017, 01:55
00:00

Difficulty:

55% (hard)

Question Stats:

68% (02:36) correct 32% (02:57) wrong based on 152 sessions

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If x and y are integers and $$72^x∗54^y=96$$, what is the value of x−y?

A. -1
B. 0
C. 1
D. 2
E. 3

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GMAT 1: 570 Q48 V22
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Re: If x and y are integers and 72^x∗54^y=96, what is the value of x−y?  [#permalink]

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07 Mar 2017, 02:12
6
1
Bunuel wrote:
If x and y are integers and $$72^x∗54^y=96$$, what is the value of x−y?

A. -1
B. 0
C. 1
D. 2
E. 3

$$72^x∗54^y=96$$

$$(3^2*2^3)^x∗(3^3*2)^y=2^5*3$$

$$3^{2x}*2^{3y}*3^{3y}*2^y=2^5*3$$

$$3^{2x+3y}*2^{3x+y}=2^5*3$$

on solving 2x+3y=1 and 3x+y=5 we get x=2 and y=-1
so x-y = 2-(-1) =3

Hence option E is correct
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Joined: 30 Jan 2018
Posts: 13
Re: If x and y are integers and 72^x∗54^y=96, what is the value of x−y?  [#permalink]

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21 Feb 2018, 14:20
1
Bunuel wrote:
If x and y are integers and $$72^x∗54^y=96$$, what is the value of x−y?

A. -1
B. 0
C. 1
D. 2
E. 3

72^x∗54^y=96 divide 6 at both side of the equation:
12^x*9^y=16 divide 3 at both side of the equation:
4^x*3^y=16/3
So x=2, y=-1
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Re: If x and y are integers and 72^x∗54^y=96, what is the value of x−y?  [#permalink]

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22 Feb 2018, 10:02
Missyy wrote:
Bunuel wrote:
If x and y are integers and $$72^x∗54^y=96$$, what is the value of x−y?

A. -1
B. 0
C. 1
D. 2
E. 3

72^x∗54^y=96 divide 6 at both side of the equation:
12^x*9^y=16 divide 3 at both side of the equation:

4^x*3^y=16/3
So x=2, y=-1

Hi Missyy

You cannot divide both sides of the exponents and use the quotients in a manner that you did.

for eg if we have $$6^2*9^2$$ and as per you approach if I divide this by 3, then I should get $$2^2*3^2=36$$

but actually it will be $$\frac{6*6*9*9}{3}=972$$.

Moreover the division is also wrong. $$72^x*54^y$$ is one number but you are individually dividing both $$72^x$$ & $$54^y$$ by 6 individually. Had there been a + or - then individual division might be possible but in case of variable exponents it is incorrect
Director
Joined: 09 Mar 2018
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Location: India
Re: If x and y are integers and 72^x∗54^y=96, what is the value of x−y?  [#permalink]

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04 Feb 2019, 23:09
Bunuel wrote:
If x and y are integers and $$72^x∗54^y=96$$, what is the value of x−y?

A. -1
B. 0
C. 1
D. 2
E. 3

Be careful in such questions

$$72^x * 54^y = 96$$

Convert the above into their factors

[2^3 * 3^2]^x * [2*3^3]^y = 2^5 * 3^1
2^3x * 2^y * 3^2x * 3^3y = 2^5 * 3^1

3x + y = 5
2x + 3y = 1

Solve them to get x =2 & y =-1

E
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Joined: 22 Sep 2018
Posts: 240
If x and y are integers and 72^x∗54^y=96, what is the value of x−y?  [#permalink]

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05 Feb 2019, 16:01
Bunuel wrote:
If x and y are integers and $$72^x∗54^y=96$$, what is the value of x−y?

A. -1
B. 0
C. 1
D. 2
E. 3

My reasoning:

72 broken down into primes is $$2^3 * 3^2$$ After multiplying x in we get $${2^3x} * 3^{2x}$$

54 broken down into primes is $$2 * 3^3$$ After multiplying y in we get $$2^y * 3^{3y}$$

96 broken down into primes is $$2^5 * 3$$

Same base means you can add the exponents together so we get: $$2^{(3x+y)} * 3^{(2x+3y)}$$

We know $$2^{(3x+y)} = 2^5$$ and $$3^{(2x+3y)} = 3^1$$

We can solve for x and y now which gives us 2 and -1 respectively. Subtract them and we get 3 (Choice E)
If x and y are integers and 72^x∗54^y=96, what is the value of x−y?   [#permalink] 05 Feb 2019, 16:01
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