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Math Expert V
Joined: 02 Sep 2009
Posts: 61189
If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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Question Stats: 36% (01:56) correct 64% (02:08) wrong based on 401 sessions

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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

Kudos for a correct solution.

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Re: If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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Bunuel wrote:
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 = 100b

Kudos for a correct solution.

Not quite sure if I got this right:

Statement 1: rewrite as b = 100/a >>> which is a*b = 100
Plug in some numbers a=2 and b=50, i.e. c > 50 in THIS case because c > b
2%*50%*55 if C is for example 55, the result will NOT be an integer. However if C is 100, the result will be integer. Therefore insufficient.

Statement 2: alone insufficent (nothing about a).

Together:
Statement 2 says that c = 100b. Combined with the information above, this will always be integer. Even if a = 1 and b = 100.

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Re: If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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Bunuel wrote:
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 = 100b

Kudos for a correct solution.

Hello,
Answer to this question will be C. Both statements combined are sufficient.

Statement 2 : C=100b not sufficient as there is no information about a. Calculating b% of c will be b/100 *100b which will give b^2. But can't be solved further as no information about a.

Statement 1 : It gives us ab=100. No information about c as well as a and b can assume different values.

Combining both Statement 1 and Statement 2 are sufficient.

b% of c = b^2
a% of b% = a/100 *b^2 => ab^2/100 => a*b*b/100 =>ab*b/100 =>100b/100 => b which is an integer.

Hence combining both statements we can say that it is an integer. Intern  Joined: 23 Apr 2014
Posts: 20
Re: If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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1
a% of b% of c = abc/(100*100)
stmt 1 - b = (a/100)^(-1) so b= 100/a substituting above we get a% of b% of c = c/100 - so not suff

stmt 2 - substituting c = 100b we get b2a/100 - not sufficient

combining 1 and 2 we get b2a/100 = 100
ans C
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Re: If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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Bunuel wrote:
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 = 100b

Kudos for a correct solution.

Question : is a% of b% of c an integer?

Question : is (a/100)*(b/100)*c and Integer?

Question : is (a*b*c)/(100*100) and Integer?

Statement 1: b = (a/100)^(-1)

i.e. b = (100/a)
i.e. ab/100 = 1 which if we substitute in the Expression of Question, we are left with c/100
but c/100 may or may not be an Integer

Hence NOT SUFFICIENT

Statement 2: c = 100b

if we substitute this value of c in the Expression of Question, we are left with a*b^2/100
but a*b^2/100 may or may not be an Integer

Hence NOT SUFFICIENT

Combining the two statements

ab/100 = 1 and c = 100b

(a*b*c)/(100*100) = (1)*(c/100) = 100b/100 = b Which is an Integer (Given)

Hence, the entire expression, (a*b*c)/(100*100) will be an INTEGER
Hence SUFFICIENT

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Math Expert V
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Posts: 61189
Re: If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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Bunuel wrote:
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

Kudos for a correct solution.

Sorry guys, it's (2) $$c = 100^b$$ NOT (2) c = 100b . Edited the original post. Please try again.
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If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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Bunuel wrote:
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

Kudos for a correct solution.

Question : is a% of b% of c an integer?

Question : is (a/100)*(b/100)*c and Integer?

Question : is (a*b*c)/(100*100) and Integer?

Statement 1: b = (a/100)^(-1)

i.e. b = (100/a)
i.e. ab/100 = 1 which if we substitute in the Expression of Question, we are left with c/100
but c/100 may or may not be an Integer

Hence NOT SUFFICIENT

Statement 2: c = 100^b

if we substitute this value of c in the Expression of Question, we are left with $$\frac{(a*b*100^b)}{(100*100)}$$
SInce a, b, and c are positive Integers and a < b < c

i.e. Minimum value of a = 1 and Minimum value of b = 2

i.e. Numerator in the expression $$\frac{(a*b*100^b)}{(100*100)}$$ is a multiple of $$100^2$$ which cancels the $$100^2$$ in denominator and renders and Integer result a multiple of $$a*b$$

Hence SUFFICIENT

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Re: If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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1
2
OA:

You can rephrase the question as follows:

Is $$\frac{a}{100}*\frac{b}{100}*c$$ an integer?

Is $$\frac{abc}{100^2}$$ an integer?

$$100^2$$ can also be thought of as $$(102)^2$$ or $$10^4$$, which in turn can be broken down into primes: $$2^45^4$$

Thus, an alternative rephrase is as follows:

Does the product abc contain four 2’s and four 5’s?

(1) INSUFFICIENT: Simplifying the statement and analyzing it with regards to divisibility, you get

ab = 100

ab = $$2^25^2$$

Thus the product ab contains two 2’s and two 5’s, but it is uncertain whether the inclusion of c in the product would add the two more 2’s and two more 5’s needed to satisfy the question.

(2) SUFFICIENT: Simplifying the statement by breaking it down into primes, you get:

c = $$(2^25^2)^b$$

This means that c contains a set of two 2’s and two 5’s, b number of times. That is, if b = 1, c contains only two 2’s and two 5’s, however, if b = 2, c contains two sets for a total of four two’s and four 5’s, etc. Looking back at the given, a < b < c and all three are positive integers, thus the minimum value for b is 2. That means that c alone must have at least four 2’s and four 5’s and therefore so will the product abc.

NOTE that this is a very tempting C-trap. Statement (1) provides information about a and b, and statement (2) provides information about c, appearing to complete the picture.

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If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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a<b<c, all integrs>0

St1. b=(a/100)^-1=100/a, so ab=100

can be 2,50; 1,100, 4,25. If c=1000000 then after all % it is integer, but if c=51,101 or 26 - non-integers. INSUFF

St2. c=100^b. The minimal b=2, a=1, then c=10000. So, 0.01*0.02*10000=2 (integer). Any higher b will give even higher integer
SUFF

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Re: If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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1
This is like one of mine favourite trick questions where C seems like the only option just because we have been in a hurry.
So, dissecting what the question is asking for we know that a<b<c and luckily they all are positive integers (Aah! lot of test cases gone )

Now : we have a situation where abc/10000= integer. In other works abc=10000n

Statement 1: ab=100
awesome but what about C. if c=100 yay we are done, but if c=17 Ouch we are not. so we have a yes no situation NS

Statement2: c=100^b
if b=1
then c=100 and a=?
oh we are messed up a little here. but hey wait doesn't the question stem say A<b<c and all of them are sweet and neat positive intergers. So b is definitely more than 1 and any value of b>1 will be sufficient to prove our case

YAY B is the correct answer.

+kudos if you like my explanation.
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Re: If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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This is a beautifully confusing question stem. At first glance this question might appear to be about percentages or inequalities, but it's really about divisibility. "Is a% of b% of c an integer" can be translated into:

(a/100) * (b/100) * (c) => abc/10000

That's a little easier. Is the product of a, b, and c divisible by 10000?

1) the product of a and b is 100, but we don't know if c is a positive multiple of 10
NOT SUFFICIENT

2) because of the constraints in the stem, we know that b is an integer greater than or equal to 2. Thus, c is an integer multiple of 100^2. Since a and be are integers, the product of a, b, and c is an integer multiple of 100^2, and is definitely divisible by 10000.
SUFFIECIENT
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Re: If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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Inference from the given data:
Question is asking whether abc = 10,000XK (where as K>0)
a<b<c (min value of a =1 => min b= 2)
1. no value about c . So Not Suff.

2. C = b power of 100.
it is given that b=2, So c= 10000 (min) & a = + ve integer.
if we substitute we will get unique answer to the given question.
So Suff.
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Re: If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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1
You can rephrase the question as follows:

Is an integer?

Is an integer?

1002 can also be thought of as (102)2 or 104, which in turn can be broken down into primes: 2454

Thus, an alternative rephrase is as follows:

Does the product abc contain four 2’s and four 5’s?

(1) INSUFFICIENT: Simplifying the statement and analyzing it with regards to divisibility, you get

ab = 100

ab = 2252

Thus the product ab contains two 2’s and two 5’s, but it is uncertain whether the inclusion of c in the product would add the two more 2’s and two more 5’s needed to satisfy the question.

(2) SUFFICIENT: Simplifying the statement by breaking it down into primes, you get:

c = (2252)b

This means that c contains a set of two 2’s and two 5’s, b number of times. That is, if b = 1, c contains only two 2’s and two 5’s, however, if b = 2, c contains two sets for a total of four two’s and four 5’s, etc. Looking back at the given, a < b < c and all three are positive integers, thus the minimum value for b is 2. That means that c alone must have at least four 2’s and four 5’s and therefore so will the product abc.

NOTE that this is a very tempting C-trap. Statement (1) provides information about a and b, and statement (2) provides information about c, appearing to complete the picture.

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Re: If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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Top Contributor
Bunuel wrote:
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

Kudos for a correct solution.

Target question: Is a% of b% of c an integer?

This is a great candidate for rephrasing the target question.
Aside: See our video with tips on rephrasing the target question (below)

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

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Re: If a, b, and c are positive integers such that a < b < c, is a% of b%  [#permalink]

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