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Post subject: GMAT Number Theory Posted: Fri Jun 18, 2010 3:40 pm 

Joined: Sun May 30, 2010 3:15 am Posts: 424

The greatest common factor of positive integers m and n is 12. What is the greatest common factor of (2m², 2n²)?
A. 2 B. 12 C. 24 D. 144 E. 288
Suppose we factorize m and n into prime factors. The greatest common factor of positive integers m and n is 12 so the only prime factors m and n have in common are 2, 2 and 3. (12 = 2 × 2 × 3).
If we factorize m² into prime factors we will get each of prime factors of m twice. The same happens to n². So the only prime factors m² and n² would have in common are 2, 2, 2, 2 and 3, 3.
If we factorize 2m² into prime factors we will get the same prime factors as for m² and one more prime factor “2”. The same happens to n². So the only prime factors 2m² and 2n² would have in common are 2, 2, 2, 2, 2 and 3, 3. By factoring those we get the greatest common factor of 2m² and 2n².
2 × 2 × 2 × 2 × 2 × 3 × 3 = 288 The answer is (E). 
Or take the lowest possible number to be 12, since 12 is a denminator of 12, hence m and n can be 12. Simply square 12 and multiply by 2 to get to 288.


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Gennadiy

Post subject: Re: math (test 4, question 16): numbers theory, common facto Posted: Fri Jun 18, 2010 4:12 pm 

Joined: Sun May 30, 2010 2:23 am Posts: 498

Yes, you are right. In this case you can do so by using assumption that the greatest common factor of (2m², 2n²) must be the same for any m or n, since we have only definite number choices. So plugging in any values of m and n that fit original statement is Ok (e.g. m = 24 and n = 32 or m = 12 and n = 12, etc.).
However if a similar question has at least one answer choice which is not a definite number, like "2m²" or "it can not be determined", then the assumption can no longer be used and by plugging in numbers you can only eliminate wrong choices. If you eliminate all choices except one then the only one that is left will be the right one.


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