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Number theory -3

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by guerrero » Fri Apr 12, 2013 8:55 am
If a and b are positive integers, and (2^3) (3^4) (5^7)=(a^3)*b, how many different possible values of b are there?
(A) 2
(B) 3
(C) 4
(D) 6
(E) 12

Source : Magoosh
OA E
Last edited by guerrero on Fri Apr 12, 2013 9:02 am, edited 1 time in total.
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by Anju@Gurome » Fri Apr 12, 2013 8:57 am
guerrero wrote:If a and b are positive integers, and (2^3)*(3^4)*(5^7) = (a^3)*b, how many different possible values of b are there?
Note that number of possible values of b will be same as the number of possible values of a^3.

The left-hand side has three 2s, four 3s, and seven 5s.

As the right-hand side contains a^3, a^3 must contain 2 and/or 3 and/or 5 in multiples of 3.
In other words a^3 can be equal to 1 or (2^3) or (2^3)*(3^3) etc.

Hence, number of possible values of a^3 = (Number of ways to select 2 in multiples of 3)*(Number of ways to select 3 in multiples of 3)*(Number of ways to select 5 in multiples of 3)

Number of ways to select 2 in multiples of 3 : In 2 ways 2^0 or 2^3
Number of ways to select 3 in multiples of 3 : In 2 ways 3^0 or 3^3
Number of ways to select 5 in multiples of 3 : In 3 ways 5^0 or 5^3 or 5^6

Hence, number of possible values of a^3 = 2*2*3 = 12

So, number of different possible values of b = 12

The correct answer is E.
Last edited by Anju@Gurome on Fri Apr 12, 2013 9:19 am, edited 1 time in total.
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by GMATGuruNY » Fri Apr 12, 2013 9:24 am
guerrero wrote:If a and b are positive integers, and (2^3) (3^4) (5^7)=(a^3)*b, how many different possible values of b are there?
(A) 2
(B) 3
(C) 4
(D) 6
(E) 12

Source : Magoosh
OA E
(2³)(3³)(5³)(5³)(3)(7) = (a³)b

a³ must be the cube of an integer
Once a value has been chosen for a³ , the value of b will be equal to whichever factors on the lefthand side are not included in the value of a³.

The answer choices represent the number of options for b.
Since the greatest answer choice is 12, there are at most 12 options for b, implying that there are at most 12 options for a³.
Thus, we can quickly count the number of options for a³:
1³
2³
3³
5³
(2³)(3³)
(2³)(5³)
(3³)(5³)

We can stop here.
Since there are at least 7 options for a³, there are at least 7 options for b.
Thus, the only viable answer choice is E.

Here are all of the options for a³ :
1³
2³
3³
5³
(2³)(3³)
(2³)(5³)
(3³)(5³)
(5³)(5³)
(2³)(3³)(5³)
(2³)(5³)(5³)
(3³)(5³)(5³)
(2³)(3³)(5³)(5³)
Total options = 12.
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