How to prove that for (a^3)*(b^2)=2000, a and b can only be 5 and 4

Question

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?

ganand · Answer

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} 	imes 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 · Answer

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?

ccooley · Answer

Correct. GMAT problems are always 'fair,' so as long as you're doing a well-written GMAT problem, you should always be able to use this technique to identify just one combination of values that works. (Unless it's Data Sufficiency!)

However, it isn't necessarily true in general. For example, the following is true:

46656 = 4^3 * 27^2

46656 = 9^3 * 8^2

How to prove that for (a^3)*(b^2)=2000, a and b can only be 5 and 4

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