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Math Revolution and GMAT Club Contest! If a and b are positive integer

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Re: Math Revolution and GMAT Club Contest! If a and b are positive integer  [#permalink]

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New post 19 Dec 2015, 03:39
n=a^3∗b^4, To find the number of factors of n.

Lets take choice 1.
(1) a and b are prime numbers

Since both a and b are prime numbers, then number of factors of n will be (3+1)(4+1)=4*5=20 factors.
This choice is sufficient.

(2) n has only prime factors 5 and 7.
Again, since both a and b are prime numbers, then number of factors of n will be (3+1)(4+1)=4*5=20 factors.
This choice is sufficient.

Each statement Alone is sufficient to answer the question and therefore choice D.
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Re: Math Revolution and GMAT Club Contest! If a and b are positive integer  [#permalink]

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New post 19 Dec 2015, 05:44
N=a^3*b^4

1) Sufficient, (3+1)*(4+1) = 20 factors

2) n has only 5, 7 as prime factors.. again we have 20 factors.. so D ans
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Re: Math Revolution and GMAT Club Contest! If a and b are positive integer  [#permalink]

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New post 20 Dec 2015, 10:45
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Bunuel wrote:

Math Revolution and GMAT Club Contest Starts!



QUESTION #10:

If a and b are positive integers, let \(n = a^3*b^4\), how many different factors n has?

(1) a and b are prime numbers
(2) n has only prime factors 5 and 7


Check conditions below:



Math Revolution and GMAT Club Contest

The Contest Starts November 28th in Quant Forum


We are happy to announce a Math Revolution and GMAT Club Contest

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MATH REVOLUTION OFFICIAL SOLUTION:

Since we have 3 variables (n, a, b) and 1 equation (\(n=a^3b^4\)) in the original condition, we need 2 equations to match the number of variables and the number of equations. Since we need both 1) and 2), the correct answer is likely C.

Using con 1) & 2), we get \(n=5^37^4\) or \(n=7^35^4\). The number of different factors is (3+1)(4+1)=20. This is unique and sufficient. Therefore, the correct answer is C.

However, since this is an “integer” question, which is one of the key questions, we should apply Common Mistake Type 4(A).

In case of con 1), if a=b, \(n=a^7\) → (7+1)=8. However, if a≠b, \(n=a^3b^4\) → (3+1)(4+1)=20. So this is not unique and sufficient.

In case of con 2), \(n=5^37^4\) → (3+1)(4+1)=20. However, \(n=1^335^4=5^47^4\) → (4+1)(4+1)=25. This is not unique and sufficient. Therefore the correct answer is C.

Note : For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: Math Revolution and GMAT Club Contest! If a and b are positive integer  [#permalink]

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New post 20 Dec 2015, 10:59
Ans is indeed C.
thanks for the great explanation Bunuel. :)
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Re: Math Revolution and GMAT Club Contest! If a and b are positive integer  [#permalink]

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New post 20 Dec 2015, 11:04
Bunuel, I think the explanation is clear on why C is correct answer, however curious on where I can get to see these error types 4(A) and 4(B) spoken about in the solution?
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Re: Math Revolution and GMAT Club Contest! If a and b are positive integer  [#permalink]

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New post 19 Jul 2018, 16:22
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Re: Math Revolution and GMAT Club Contest! If a and b are positive integer &nbs [#permalink] 19 Jul 2018, 16:22

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