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

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