If x is a positive integer, is (10x - 1) divisible by the positive int

Solution

Step 1: Analyse Question Stem

• x and y are positive integers.
• We need to find, if \((10x-1)\) is divisible by y

o Or, if \(\frac{(10x – 1)}{y }= k\) , where k is positive integer.
o Or, if \(\frac{10x}{y }-\frac{1}{y} = k\)

 Hence, for k to be a positive integer either both \(\frac{10x}{y}\) and \(\frac{1}{y}\) have to be integers or both have to be fractions.

Step 2: Analyse Statements Independently

• According to this statement, \(\frac{x}{y} \) is an integer

o \(\frac{10x}{y }\) is an integer
• However, we do not know whether of \(\frac{1}{y}\) is an integer or not.

o Let’s take two simple cases to understand the above statement:

 Case 1: If \(y = 1\) then \(\frac{1}{y }=1\) is an integer
 Case 2: If \(y ≠ 1\) then \(\frac{1}{y}\) is not an integer.
 The results of above two cases are not same. Therefore, we cannot conclude about the nature of 1/y.

• According to this statement, \(y ≠ 1 \)

o \(\frac{1}{y }\)is a fraction
• However, we do not know whether \(\frac{10x}{y }\)is integer or not.

Step 3: Analyse Statements by combining.

• According to statement 1: \(\frac{10x}{y }\)is an integer
• According to statement 2: \(\frac{1}{y} \)is a fraction
• On combining both, we get

o \(\frac{10x}{y} - \frac{1}{y }\)is not an integer i.e. \(\frac{10x}{y} - \frac{1}{y }≠ k\)