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If x is a positive integer, is x – 1 a factor of 104? (1) x is divisible by 3. (2) 27 is divisible by x.

For me the answer is clearly B. But OA is C. Can someone please explain?

If x is a positive integer, is x – 1 a factor of 104?

(1) x is divisible by 3 --> well, this one is clearly insufficient, as x can be 3, x-1=2 and the answer would be YES but if x is 3,000 then the answer would be NO.

(2) 27 is divisible by x --> factors of 27 are: 1, 3, 9, and 27. Now, if x is 3, 9, or 27 then the answer would be YES (as 2, 8, and 26 are factors of 104) BUT if x=1 then x-1=0 and zero is not a factor of ANY integer (zero is a multiple of every integer except zero itself and factor of none of the integer). Not sufficient.

(1)+(2) As from (1) x is a multiple of 3 then taking into account (2) it can only be 3, 9, or 27. For all these values x-1 is a factor of 104. Sufficient.

The question deals with the concepts of factors and multiples of a number. Its important to analyze the information given in the question first before preceding to the statements. Please find below the detailed solution:

Step-I: Understanding the Question The question tells us that \(x\) is a positive integer and asks us to find if \(x-1\) is a factor of 104

Step-II: Draw Inferences from the question statement Since \(x\) is a +ve integer, we can write \(x>0\). The question talks about the factors of 104. Let's list out the factors of 104.

\(104 = 13 * 2^3\). So, factors of 104 are {1,2,4,8,13,26,52,104}, a total of 8 factors.

If \(x-1\) is to be a factor of 104, \(2<=x<=105\). With these constraints in mind lets move ahead to the analysis of the statements.

Step-III: Analyze Statement-I independently St-I tells us that \(x\) is divisible by 3. This would mean that \(x\) can take a value of any multiple of 3. Now, all the multiples of 3 are not factors of 104. So, we can't say for sure if \(x-1\) is a factor of 104. Hence, statement-I alone is not sufficient to answer the question.

Step-IV: Analyze Statement-II independently St-II tells us that 27 is divisible by \(x\) i.e. \(x\) is a factor of 27. Let's list out the factors of 27 - {1,3,9,27}. But, we know that for \(x-1\) to be a factor of 104, \(2<=x<=105\). We see from the values of factors of 27, \(x\) can either be less than 2(i.e. 1) or greater than 2 (i.e. 3,9 & 27). Hence, statement-II alone is not sufficient to answer the question.

Step-V: Analyze both statements together St-I tells us that \(x\) is a multiple of 3 and St-II tells us that \(x\) can take a value of {1, 3, 9, 27}. Combining these 2 statements we can eliminate \(x=1\) from the values which \(x\) can take. So, \(x\)={ 3, 9, 27} and \(x-1\) = {2, 8, 26}. We observe that all the values which \(x-1\) can take is a factor of 104. Hence, combining st-I & II is sufficient to answer our question.

Answer: Option C

Takeaway Analyze the information given in the question statement properly before proceeding for analysis of the statements. Had we not put constraints on the values of x, we would not have been able to eliminate x=1 from st-II analysis.

If x is a positive integer, is x-1 a factor of 104? [#permalink]

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21 Sep 2016, 00:47

prashantrchawla wrote:

Let's take x=30, in this case, 1. A Will be sufficient. However 30-1 is 29 is not a factor of 104. 2. 27 is also not divisible by 30. Not sufficient.

Hence, In this case is the answer E. Can anybody answer my doubt.

I am not sure what you are trying to do here.

for statement 1 : you are considering only one value of x, which is making your case sufficient. Take x =3 and x = 6, you will get 104 divisible for x-1 = 2 but not for x-1 = 5.

Hence, it is Insufficient.

Statement 2 : We are given 27 is divisible by x. It means x is a factor of 27. The factors could be 1,3,9 and 27.

Divide 104 by each of (x-1) as 0, 2,8 and 26. You will find 104 divisible by all but 0. hence, insufficient.

On combining, we know that x cannot be 0. Hence, Answer C.
_________________

Let's take x=30, in this case, 1. A Will be sufficient. However 30-1 is 29 is not a factor of 104. 2. 27 is also not divisible by 30. Not sufficient.

Hence, In this case is the answer E. Can anybody answer my doubt.

Your logic there is not clear. Why do you take x as 30? You cannot arbitrarily take x to be 30 and work with this value only. Also, how is the first statement sufficient? If x is 3, then x-1=2 and the answer would be YES but if x is 3,000 then the answer would be NO.

Please re-read the solutions above.
_________________

Re: If x is a positive integer, is x-1 a factor of 104? [#permalink]

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04 Dec 2017, 10:54

enigma123 wrote:

If x is a positive integer, is x – 1 a factor of 104?

(1) x is divisible by 3. (2) 27 is divisible by x.

Lets look at the prime factorisation of 104: 2^3 * 13 Thus factors of 104 = 1, 2, 4, 8, 13, 26, 52, 104 We are asked whether x-1 is one of these 8 integers, OR IS x one of these: 2, 3, 5, 9, 14, 27, 53, 105

(1) x is divisible by 3, so x could be any multiple of 3 like 9 or 27 or 54. Insufficient.

(2) 27 is divisible by x, so x is a factor of 27. Now factors of 27 are: 1, 3, 9, 27. If x is 1, then x-1 is 0 and thus NOT a factor of 104, but if x is 3 or 9 or 27, then x-1 will take values as 2 or 8 or 26 respectively, and thus BE a factor of 104. So Insufficient.

Combining the two statements, x has to be a multiple of 3, yet a factor of 27 also. So x could be either 3 or 9 or 27. For each of these cases, x-1 will be a factor of 104, as explained in statement 2. Sufficient. Hence C answer