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Cez005
Joined: 13 Dec 2013
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GMAT 1: 710 Q46 V41
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28 Apr 2017, 21:15
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If (a^3)*(b^2)=2000, and a and b are integers, can anyone shed light on how to prove that a and b can only be 5 and 4 respectively?
(5^3)*(4^2)=2000, but how do we know that these are the only possible integer values? Prime factorization of 2000?
(5^3)*(4^2)=2000, but how do we know that these are the only possible integer values? Prime factorization of 2000?
28 Apr 2017, 21:45
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Cez005
If (a^3)*(b^2)=2000, and a and b are integers, can anyone shed light on how to prove that a and b can only be 5 and 4 respectively?
(5^3)*(4^2)=2000, but how do we know that these are the only possible integer values? Prime factorization of 2000?
(5^3)*(4^2)=2000, but how do we know that these are the only possible integer values? Prime factorization of 2000?
Hi Cez005,
Any integer greater than 1 either is prime itself or is the product of prime numbers and that product is unique.
Above result is known as the Fundamental theorem of arithmetic.
So, if you do the prime factorization of 2000, we have \(2000 = 5^{3} \times 4^{2}\). By the fundamental theorem of arithmetic, we know that this is unique.
Proof of the theorem is a bit involved and not required for GMAT.
However, if you are interested in learning the proof then refer the wiki page.
Fundamental_theorem_of_arithmetic
Hope it helps.
Cez005
Joined: 13 Dec 2013
Last visit: 11 Feb 2020
Posts: 100
Own Kudos:
Given Kudos: 122
Location: United States (NY)
Concentration: General Management, International Business
Schools: Cambridge"19 (A)
GMAT 1: 710 Q46 V41
GMAT 2: 720 Q48 V40
GPA: 4
WE:Consulting (Consulting)
Products:
28 Apr 2017, 22:34
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ganand
Cez005
If (a^3)*(b^2)=2000, and a and b are integers, can anyone shed light on how to prove that a and b can only be 5 and 4 respectively?
(5^3)*(4^2)=2000, but how do we know that these are the only possible integer values? Prime factorization of 2000?
(5^3)*(4^2)=2000, but how do we know that these are the only possible integer values? Prime factorization of 2000?
Hi Cez005,
Any integer greater than 1 either is prime itself or is the product of prime numbers and that product is unique.
Above result is known as the Fundamental theorem of arithmetic.
So, if you do the prime factorization of 2000, we have \(2000 = 5^{3} \times 4^{2}\). By the fundamental theorem of arithmetic, we know that this is unique.
Proof of the theorem is a bit involved and not required for GMAT.
However, if you are interested in learning the proof then refer the wiki page.
Fundamental_theorem_of_arithmetic
Hope it helps.
This is helpful, thanks a lot. Through prime factorization you can identify the bases/primes and from there it should always be possible to identify the base/power combinations, correct?
