Question-5

Let \(p_1\) and \(p_2\) denote two arbitrary prime numbers. Which one of the following statements is correct for all values of \(p_1\) and \(p_2\)?

• \(p_1 + p_2\) is not a prime number
• \(p_1p_2\) is not a prime number
• \(p_1 + p_2 + 1\) is a prime number
• \(p_1p_2 + 1\) is a prime number

NoteAnswer

• \(p_1 + p_2\) is not a prime number
• \(p_1p_2\) is not a prime number
• \(p_1 + p_2 + 1\) is a prime number
• \(p_1p_2 + 1\) is a prime number

NoteSoluton

• If \(\displaystyle p_{1} =2,p_{2} =7\), then \(\displaystyle p_{1} +p_{2} =9\) is not prime. However, if \(\displaystyle p_{2} =2,p_{2} =3\), then \(\displaystyle p_{1} +p_{2} =5\) is a prime.

• \(\displaystyle p_{1} p_{2}\) has two prime factors other than itself and \(\displaystyle 1\). Hence \(\displaystyle p_{1} p_{2}\) is not prime for all primes \(\displaystyle p_{1}\) and \(\displaystyle p_{2}\).

• If \(\displaystyle p_{1} =2,p_{2} =3\), then \(\displaystyle p_{1} +p_{2} +1=6\) is not a prime.

• If \(\displaystyle p_{1} =3,p_{2} =5\), then \(\displaystyle p_{1} p_{2} +1=16\) is not a prime.