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If a, b, and c are positive integers such that a < b <

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If a, b, and c are positive integers such that a < b <

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Source: Manhattan Prep

If a, b, and c are positive integers such that a < b < c, is a% of b% of c an integer?

1) b = (a/100)^-1
2) c = 100^b

The OA is B

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Quote:
If a, b, and c are positive integers such that a < b < c, is a% of b% of c an integer?

(1)b = (a/100)^-1
(2) c = 100^b
Target question: Is a% of b% of c an integer?

This is a great candidate for rephrasing the target question.
Aside: We have a free video with tips on rephrasing the target question: http://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1100

a% of b% of c is the same as (a/100)(b/100)(c), which equals abc/10,000
So, we can rephrase the target question as follows:
REPHRASED target question: Is abc/10,000 an integer?

We can REPHRASE the target question even further...
RE-REPHRASED target question: Is abc a multiple of 10,000?

Statement 1: b = (a/100)^-1
In other words, b = 100/a
There are several values of a, b and c that satisfy this condition. Here are two:
Case a: a = 1, b = 100 and c = 1000, in which case abc = 100,000. Here, abc IS a multiple of 10,000
Case b: a = 1, b = 100 and c = 101, in which case abc = 10,100. Here, abc is NOT a multiple of 10,000
Since we cannot answer the RE-REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: c = 100^b
IMPORTANT: We are told that a, b and c are POSITIVE INTEGERS and that a < b < c
So, we can be certain that b > 2.
If b is greater than or equal to 2, then c (which equals 100^b) can equal 10,000 or 1,000,000 or 100,000,000 and so on.
Notice that ALL of these possible values of c are multiples of 10,000
So, if c is a multiple of 10,000, then abc MUST be a multiple of 10,000
Since we can answer the RE-REPHRASED target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

For even more information on rephrasing the target question, you can read this article I wrote for BTG: http://www.beatthegmat.com/mba/2014/06/30/rephrasing-the-target-question

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BTGmoderatorLU wrote:
Source: Manhattan Prep

If a, b, and c are positive integers such that a < b < c, is a% of b% of c an integer?

1) b = (a/100)^-1
2) c = 100^b
Does (a/100) * (b/100) * c = integer?

Test integer values such that a < b < c.

Statement 1: b = (a/100)¯¹
Thus, b = 100/a.

Test the smallest possible value for a.
Case 1: a=1
Here, b =100/1 = 100.
In this case, (a/100) * (b/100) * c = 1/100 * 100/100 * c = c/100.

If c = 200, then c/100 = 2, which is an integer.
If c = 201, then c/100 = 201/100, which is not an integer.
INSUFFICIENT.

Statement 2: c = 100^b
Test the smallest possible value for b.
Case 2: b=2
Here, a=1 and c = 100² = 10000.
In this case, (a/100) * (b/100) * c = 1/100 * 2/100 * 10000 = 2, which is an integer.

Test an extreme value for b.
Case 3: b=100
Here, c = 100¹⁰⁰.
In this case, (a/100) * (b/100) * c = a/100 * 10/100 * 100¹⁰⁰ = 10a * 100⁹⁸, which is an integer.

Cases 2 and 3 illustrate that -- given that c = 100^b -- (a/100) * (b/100) * c will always be equal to an integer value.
SUFFICIENT.

The correct answer is B.

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BTGmoderatorLU wrote:
Source: Manhattan Prep

If a, b, and c are positive integers such that a < b < c, is a% of b% of c an integer?

1) b = (a/100)^-1
2) c = 100^b
$$1 \le a < b < c\,\,\,{\rm{ints}}\,\,\,\left( * \right)$$
$$\left[ {{a \over {100}}\left( {{b \over {100}}\left( c \right)} \right) = } \right]\,\,\,\,\,{{abc} \over {{{10}^4}}}\,\,\mathop = \limits^? \,\,{\mathop{\rm int}} $$

$$\left( 1 \right)\,\,b = {{100} \over a}\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,100,1000} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,100,101} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$

$$\left( 2 \right)\,\,c = {\left( {{{10}^2}} \right)^b}\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,c = {\left( {{{10}^2}} \right)^{ \ge \,\,2}} = {10^{ \ge \,4}}$$
$${{abc} \over {{{10}^4}}} = {{ab \cdot {{10}^{ \ge \,4}}} \over {{{10}^4}}}\,\,\mathop = \limits^{\left( * \right)} \,\,{\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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