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Abe, Beth, Carl and Duncan are four siblings, among which Abe and Carl are twins and Beth and Duncan are also twins. When the present ages of the four siblings are multiplied, the product is 900. If Beth is older than Abe, what is the age of Duncan? Assume the ages of all siblings to be integers.

(1) The difference between Beth’s age and Abe’s age is a prime number.

(2) If Carl had been born four years earlier, the difference between Duncan’s age and Carl’s age would have been a prime number.

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Re: Abe, Beth, Carl and Duncan are four siblings, among which Abe and Carl [#permalink]

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10 Apr 2015, 08:38

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Let the ages be A,B,C & D respectively. Given A=C & B=D, A*B*C*D=900=>B^2*A^2=900=>B*A=30. Possible combinations are (30*1),(15*2),(10*3),(6*5) Statement 1: B-A=P(prime) => (30*1),(15*2), & (10*3) satisfy the equation. Therefore statement 1 is insufficient. Statement 2: D-(C+4)=P or B-(A+4)=P or B-A-4=P. Now only (10*3) satisfy. Therefore B=D=10 and statement 2 is sufficient. Answer: (B)
_________________

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Re: Abe, Beth, Carl and Duncan are four siblings, among which Abe and Carl [#permalink]

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10 Apr 2015, 09:56

Shuvabrata88,

I agree with the approach which you have taken but I think the answer should be E as statement 2 simply states that diff b/w D and C is prime so it could be either D-C-4 or C+4-D; in this case (5*6) combination also works out to be prime.

Initial ques stem mentions that B>A i.e D>C but if we add 4 to C then this statement may not hold true.

I may be over complicating the question, lets wait for OE

We are told about four siblings Abe, Beth, Carl and Duncan such that Abe and Carl are twins and Beth and Duncan are also twins. We are also given that the product of the present ages of the four siblings is 900. Further we are told that Beth is older than Abe and we are asked to find the age of Duncan

Step-II: Interpreting the Question Statement

Since Abe and Carl are twins, their ages would be same, let’s assume it to be \(x\). Similarly, since Beth and Duncan are twins, their ages would be same, let’s assume it to be \(y\).

We are told that Beth is older than Abe, i.e. \(y > x\) and the product of the ages of the siblings is 900, so we can write \(x^ 2 * y^2 = 900\).

We can observe here that 900 is written as product of two squares, since 900 can be prime factorized as \(900 = 2^2 * 3^2 * 5^2\), the possible set of values of (\(x, y\)) can be:

• (1, 30) or • (2, 15) or • (3, 10) or • (5,6)

Let’s proceed to the solutions to see if we can get a unique value of \(x\) with this understanding.

Step-III: Statement I

Statement tells us that \(y\) \(–\) \(x\) is a prime number. Let’s evaluate our possible cases to see if we can find a unique value for \(x\):

• (1, 30) –>= 29-> Prime • (2, 15) –>= 13 -> Prime • (3, 10) – > = 7 -> Prime • (5,6) – > = 1 -> Not Prime

We observe here that, there are three possible values for \(x\), hence statement-I is not sufficient to arrive at the answer.

Step-IV: Statement II

Statement-II tells us that had Carl been born four years earlier, the difference between Duncan’s age and Carl’s age would have been a prime number. Since Carl’s present age is \(x\), had he been born four years earlier, his present age would be (\(x +4\)). The Statement tells us that \(y\) \(–\) \((x +4)\) is a prime number. Let’s evaluate our possible cases to see if we can find a unique value for \(x\).

• (1, 30) –>= 25-> Not Prime • (2, 15) –>= 9 -> Not Prime • (3, 10) – > = 3 -> Prime • (5,6) – > = 3 -> Prime

We observe here that there are two possible values for \(x\), hence statement-II is not sufficient to arrive at the answer.

Step-V: Combining Statements I & II

Statement-I gives us the possible values of (\(x, y\)) as (1, 30), (2, 15) and (3, 10). Statement-II gives us the possible values of (\(x, y\)) as (3, 10) and (5,6).

Combining statement-I & II give us only possible option for values of (\(x, y\)) which is (3, 10). Thus combination of St-I & II is sufficient to answer the question. Hence, the correct answer is Option C

Key Takeaways

1. Prime factorize a number to understand the ways in which a number can be represented

atom - you were right in describing why Statement-II alone is not sufficient but you did not consider the combinations of statement- I & II.

i am sorry but i disagree with your evaluation of statement two. following are my points of dispute: It is given that beth ( & hence duncan, since they are twins) is older than abe ( & correspondingly carl). That is a fact and cannot be challenged, right ? Also, fact 2: clearly states difference between Duncan's age and Carl's age..makes sense because we would want to end up with negetive number which would occur if the reverse ( C - D) were to happen..so far so good. now consider the possible age value sets (keeping in mind the logical validity of the statement )

" Since Carl’s present age is x, had he been born four years earlier, his present age would be (x +4). The Statement tells us that y – (x +4) is a prime number. " this is your explanation..totally with you till this point. HOWEVER,

y – (x +4) when applied to the set (5,6) yields -3 (which is not positive first of all forget being prime) and not 3 as you said..absolutely illogical and hence not a valid set i feel. the other sets (resultant-prime or otherwise) at least yield a positive number since x and x+4 are both lesser than y. And if we were to take this into account, the only valid set is (3,10) from stmnt 2..sufficient ans: B

WOULD REQUEST your thoughts on this and would kindly appreciate if A MATH EXPERT would care to enlighten or clarify.

Re: Abe, Beth, Carl and Duncan are four siblings, among which Abe and Carl [#permalink]

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12 Apr 2015, 11:12

smashbiker84,

I agree that D>C is given in the question, but how would you know D>C+4?? we can have C+4>D right? Hence we must need to consider D-C-4 and C+4-D; in this case 5,6 gives you the prime output

Re: Abe, Beth, Carl and Duncan are four siblings, among which Abe and Carl [#permalink]

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12 Apr 2015, 20:04

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This is my question. Well we can come up with the following pairs (A,B) or (D, C) = (15, 2), (6, 5), (10, 3), or (30, 1). From statement 2 we get that had Carl been born four years earlier, the difference would have been a prime number. So from the given information we know that Carl's age is either 2,5,3,1. So if he was born 4 years earlier then his age could have been (-2, 1, -1, -3). Well obviously only 1 is possible, so the difference between 6 and 1 be a prime number.

But clearly that is not the way to approach this problem, nor is this approach taken by anyone. I just want to know what am I missing here.

Hi smashbiker84 We are not challenging any facts here. Statement-II presents a 'What If' scenario which gives another relation between the ages of the siblings. The question statement talks about the present situation, while statement-II talks about an alternate situation. For example, assume a situation where you are 3 years elder to your sister. If your sister were to be born 4 years earlier, she would become a year elder to you. This is not contradicting the present situation, but building upon the present situation.

To answer your 2nd question, please note that difference between two things is positive in real world scenarios. For example, difference between a son's age and his father's age is 30 years and the difference between the father’s and his son’s age is also 30 years. We do not say that the difference between son’s age and father’s age is -30 years. The "-" is only an indication that you have subtracted the larger item from the smaller item. Hence, when we are talking about difference between two things, it would mean the absolute difference or the magnitude of the difference. .

Hi nphatak, if a person is born earlier, his age would increase, so instead of subtracting 4 from the present age, you would need to add 4 in the present age. For ex: If a person is born in 2004, his present age would be 11, but if he is born four years earlier i.e. in 2000, his present age would be (11 + 4) = 15. So, in this case Carl's age would be (6,9,7,5) instead of (-2, 1, -1, -3).

Re: Abe, Beth, Carl and Duncan are four siblings, among which Abe and Carl [#permalink]

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20 Apr 2017, 02:25

Option C

In terms of age, A = C & B = D. Also, A*B*C*D = 900 =A^2*B^2 = C^2*D^2

A*B = C*D = 30

Which means, (A,B) or (B,A) could possibly be (1,30) ; (2,15) ; (3,10) ; (5,6).

I: B - A = Prime number. Let us look at the possible options: B - A = |30 -1| = 29 or |15 - 2| = 13 or |10 - 3| = 7 or |6 - 5| = 1.

Except for the last combination (6,5), rest satisfy the condition of being Prime number, so there is NO unique solution. Hence, I is Insufficient.

II: Carl's age today, had he born four years ago, would be C +4. So, D - (C + 4) = B - (A+4) = Prime number. Now again looking at the possible combinations, we can see:

30 - (1+4) = 25 Not a Prime 15 - (2+4) = 9 Not a Prime 10 - (3+4) = 3 Prime

Again, no Unique solution. Insufficient.

Now, if we combine both I + II: I:B-A = Prime Number Also, II:B - (A+4) = Prime Number From possible solutions, only one combination B = 10 & A = 3 satisfy both the conditions. As, B-A = 10 -3 = 7 Prime B - (A+4) = 10 - (3+4) = 3 Prime Hence, Sufficient.
_________________

Re: Abe, Beth, Carl and Duncan are four siblings, among which Abe and Carl [#permalink]

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31 Oct 2017, 14:38

EgmatQuantExpert wrote:

Detailed Solution

Step-I: Given Info

We are told about four siblings Abe, Beth, Carl and Duncan such that Abe and Carl are twins and Beth and Duncan are also twins. We are also given that the product of the present ages of the four siblings is 900. Further we are told that Beth is older than Abe and we are asked to find the age of Duncan

Step-II: Interpreting the Question Statement

Since Abe and Carl are twins, their ages would be same, let’s assume it to be \(x\). Similarly, since Beth and Duncan are twins, their ages would be same, let’s assume it to be \(y\).

We are told that Beth is older than Abe, i.e. \(y > x\) and the product of the ages of the siblings is 900, so we can write \(x^ 2 * y^2 = 900\).

We can observe here that 900 is written as product of two squares, since 900 can be prime factorized as \(900 = 2^2 * 3^2 * 5^2\), the possible set of values of (\(x, y\)) can be:

• (1, 30) or • (2, 15) or • (3, 10) or • (5,6)

Let’s proceed to the solutions to see if we can get a unique value of \(x\) with this understanding.

Step-III: Statement I

Statement tells us that \(y\) \(–\) \(x\) is a prime number. Let’s evaluate our possible cases to see if we can find a unique value for \(x\):

• (1, 30) –>= 29-> Prime • (2, 15) –>= 13 -> Prime • (3, 10) – > = 7 -> Prime • (5,6) – > = 1 -> Not Prime

We observe here that, there are three possible values for \(x\), hence statement-I is not sufficient to arrive at the answer.

Step-IV: Statement II

Statement-II tells us that had Carl been born four years earlier, the difference between Duncan’s age and Carl’s age would have been a prime number. Since Carl’s present age is \(x\), had he been born four years earlier, his present age would be (\(x +4\)). The Statement tells us that \(y\) \(–\) \((x +4)\) is a prime number. Let’s evaluate our possible cases to see if we can find a unique value for \(x\).

• (1, 30) –>= 25-> Not Prime • (2, 15) –>= 9 -> Not Prime • (3, 10) – > = 3 -> Prime • (5,6) – > = 3 -> Prime

We observe here that there are two possible values for \(x\), hence statement-II is not sufficient to arrive at the answer.

Step-V: Combining Statements I & II

Statement-I gives us the possible values of (\(x, y\)) as (1, 30), (2, 15) and (3, 10). Statement-II gives us the possible values of (\(x, y\)) as (3, 10) and (5,6).

Combining statement-I & II give us only possible option for values of (\(x, y\)) which is (3, 10). Thus combination of St-I & II is sufficient to answer the question. Hence, the correct answer is Option C

Key Takeaways

1. Prime factorize a number to understand the ways in which a number can be represented

Regards Harsh

This interpretation of between of in the problem seems to go against what is used on GMAT--"difference between x and y" is interpreted as "x-y"--this is also shown in GMAT club math book; am I missing something here because the difference has a direction associated with it (+/-) so (5 and 6) would actually result in (6-(5+4)) [representing Duncan - Carl Born Four Years Ealier] is actually negative and thus cannot be a prime so statement II alone should be sufficient and the right answer should be B...