What is the remainder when a positive integer P is divided by 6?

Solution

Step 1: Analyse Question Stem

• P is a positive integer.
• We need to find the remainder when P is divided by 6 or \((\frac{P}{6})_{remainder}\)

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE

• According to this statement: \(P = 12n + 4\), where n is a positive integer

o So, \((\frac{P}{6}) = 2n + \frac{4}{6}\)
• Hence, \((\frac{P}{6})_{remainder} = (\frac{4}{6})_{remainder} = 4\)

• According to this statement: \(P = 3m + 1\), where m is a positive integer. Here m can be an even integer or odd integer, so there will be two cases as given below:

o Case 1: m is an even integer i.e. \(m = 2k\) where k is any integer.

 \(P = 3*2k + 1\)
 \((\frac{P}{6}) = k + \frac{1}{6}\)
 Hence, \((\frac{P}{6})_{remainder} = (\frac{1}{6})_{remainder} = 1\)
o Case 2: m is an odd integer. Let say \(m = 2k +1\), where k is a positive integer.

 \(P = 3*(2k+1) + 1 = 6k +4\)
 \((\frac{P}{6}) = k + \frac{4}{6}\)
 Hence, \((\frac{P}{6})_{remainder} = (\frac{4}{6})_{remainder} = 4\)