What is the value of x+y?

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CappyAA: Master | Next Rank: 500 Posts; Posts: 124; Joined: Mon Jun 16, 2008 11:07 amFollowed by: 14 members; Thanked: 21 times; GMAT Score:750

by CappyAA » Tue Aug 12, 2008 5:19 pm

If x and y are positive integers, what is the value of x+y?

(1) (2^x)(3^y) = 72
(2) (2^x)(2^y) = 32

Thanks,
Cappy

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aspire750: Junior | Next Rank: 30 Posts; Posts: 16; Joined: Sun Jul 27, 2008 9:45 pm; Thanked: 2 times

Re: What is the value of x+y?

by aspire750 » Tue Aug 12, 2008 11:21 pm

CappyAA wrote:If x and y are positive integers, what is the value of x+y?

(1) (2^x)(3^y) = 72
(2) (2^x)(2^y) = 32

Thanks,
Cappy

Stm1. 2^3*3^2=72
x+y=5 Sufficient

Stm2. 2^2*2^3=32
x+y=5 Sufficient
D

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foobarnull: Senior | Next Rank: 100 Posts; Posts: 38; Joined: Wed Jul 30, 2008 9:16 pm

Re: What is the value of x+y?

by foobarnull » Mon Aug 25, 2008 4:37 pm

aspire750 wrote: Stm1. 2^3*3^2=72
x+y=5 Sufficient

Stm2. 2^2*2^3=32
x+y=5 Sufficient
D

Thanks for this. I get the exponents completely but I'm... wondering what's the arithmetic work behind this? I wasn't aware that there's an operation for multiplying different exponents on different bases. Please help... thanks.

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

Re: What is the value of x+y?

by Ian Stewart » Mon Aug 25, 2008 6:33 pm

foobarnull wrote:
aspire750 wrote: Stm1. 2^3*3^2=72
x+y=5 Sufficient

Stm2. 2^2*2^3=32
x+y=5 Sufficient
D
Thanks for this. I get the exponents completely but I'm... wondering what's the arithmetic work behind this? I wasn't aware that there's an operation for multiplying different exponents on different bases. Please help... thanks.

Ah, no- you're right, and there is no rule for 'multiplying exponents on different bases'. aspire750 was just using prime factorizations to solve the problem. If you look, for example, at statement 1), we have:

(2^x)(3^y) = 72

We know x and y are positive integers. Prime factorize 72:

72 = (2^3)*(3^2)

so we know

(2^x)(3^y) = (2^3)*(3^2)

Since x and y are positive integers, the left side and the right side of this equation are both prime factorizations. By the aptly named 'Fundamental Theorem of Arithmetic', prime factorizations are unique. That is, if one prime factorization equals another, the powers on each prime must be the same. If

(2^x)(3^y) = (2^3)*(3^2)

then the power on 2 on the left side must be equal to the power on 2 on the right side: x must be equal to 3. Same with the power on 3: y must be 2. Since we know x and y, we can add them: x+y = 5.

The second statement is a bit easier (though I'd note that, using Statement 2 alone, you cannot conclude as aspire750 did that x = 3 and y = 2; there are other possibilities):

(2^x)(2^y) = 32
2^(x+y) = 2^5
x+y = 5

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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