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

Question

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.

Bunuel · Answer

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 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.

Answer: C.

enigma123 · Answer

Thanks very much buddy for shedding light on concept of ZERO.

EgmatQuantExpert · Answer

Hi Guys,

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  .

Hope it helps!

Regards
Harsh

prashantrchawla · Answer

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.

abhimahna · Answer

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.

Bunuel · Answer

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.

amanvermagmat · Answer

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

NikhilJose · Answer

Hi  Bunuel ,
Should we consider division by zero for the GMAT?

I disregarded 1 for this question and hence got B.

Thanks in advance for your response.

Nik

Bunuel · Answer

We are told that x is a  positive  integer, so x cannot be 0.

Kinshook · Answer

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

104 = 2*2*2*13 = 2^3*13

(1) x is divisible by 3. 
If x = 3; x-1 = 2 is a factor of 104
But if x=6; x-1 = 5 is NOT a factor of 104
NOT SUFFICIENT

(2) 27 is divisible by x.
27 = 3^3 
x = {1,3,9,27}
x - 1 = {0,2,8,26}
27 is divisible by {2,8,26} but NOT divisible by 0.
NOT SUFFICIENT

(1) + (2)
(1) x is divisible by 3. 
(2) 27 is divisible by x.
x = {3,9,27}
x - 1 = {2,8,26}
104 is divisible by {2,8,26}
SUFFICIENT

IMO C

Kriti1055 · Answer

Hi  Kinshook ,  Bunuel

I am a little confused here. Request your help.

Should we consider 0 (where x=1) a factor since anything divided by zero is undefined?

Knowing that 0 cannot be a factor of any number, is it not possible to automatically discard 1 as of the possibilities and keep the remaining-27,3,9, making St. B sufficient to be the correct answer?

Bunuel · Answer

(2) says:  27 is divisible by x . So, x can be 1, 3, 9, and 27.

The question asks:  is x – 1 a factor of 104?

If x = 1, then the question becomes:  is 0 a factor of 104?  The answer to this question is NO (0 is NOT a factor of 104).

So, there is no problem for x to be 1, it just gives a NO answer to the question.

Hope it helps.

KarishmaB · Answer

104 = 2 * 2 * 2 * 13 So factors of 104 would be combinations of these... 1, 2, 4, 8, 26, etc For (x-1) to be a factor of 104, it should be one of these. (1) x is divisible by 3. x could be 3 so (x - 1) would be 2 - a factor of 104 x could be 6 so (x-1) would be 5 - not a factor of 104 Not sufficient (2) 27 is divisible by x. So x can be only 1 or 3 or 9 or 27 x could be 1 so (x - 1) = 0 - not a factor of 104 x could be 3 so (x - 1) = 2 - a factor of 104 Not Sufficient Using both statements, we see that x can be 3 or 9 or 27 only. So (x-1) will be 2 or 8 or 26. Each one of these is a factor of 104. Hence, sufficient. (x-1) is a factor of 104. Answer (C) For more on factors, check this video and these posts: https://youtu.be/DxIH8rjhpKY https://anaprep.com/number-properties-f ... -a-number/ https://anaprep.com/number-properties-r ... e-factors/

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

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