pq < 70, where both P and Q are distinct odd primes. Determine PQ.

Solution

Step 1: Analyse Question Stem

• \(p*q < 70\)

o p and q are distinct odd primes

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

• \(PQ = 2^x+1\), for some integer x
• \(2^x + 1 < 70\)

o There are two possible values of x

 Case 1: If \(x = 5\), then \(2^5 + 1 = 33 = 3*11 < 70\)

• 11 and 5 both are odd primes
 Case 2: If \(x = 6\), then \(2^6 + 1 = 65 = 5*13\)

• 5 and 13 both are odd primes.
o We are not getting unique answer.

• \(PQ < 70\)

o Case 1: If \(p = 7\) and \(q = 3\), p and q are odd primes and \(pq=21 < 70\)

 \(2+1 = 3\), which is a prime number.
o Case 2: If \(p = 5\) and \(q = 13\), p and q are odd primes and pq =65 < 70

 \(6 + 5 = 11\), which is a prime number.
• We are not getting unique answer.

Step 3: Analyse Statements by combining.

• If \(pq = 33\), then \(3 + 3 = 6\), which is not a prime number
• If \(pq = 65\), then \(6 + 5 = 11\), which is a prime number

o \(pq = 65\)