Question

Let $p$, $q$ and $r$ be three distinct prime numbers. Check whether $pqr + q$ is a composite number or not. Further, give an example for three distinct primes $p$, $q$, $r$ such that
(i) $pqr + 1$ is a composite number
(ii) $pqr + 1$ is a prime number

Hint:

Try small primes and compute $pqr \pm 1$ to test primality or compositeness.

Answer 1

Let $p = 2, q = 3, r = 5$
Then $pqr + q = 2 \cdot 3 \cdot 5 + 3 = 30 + 3 = 33$ → Composite
(i) $pqr + 1 = 30 + 1 = 31$ → Prime
Try $p = 2, q = 3, r = 7 \Rightarrow pqr = 42$
Then $pqr + 1 = 43$ → Prime again
Try $p = 2, q = 5, r = 7 \Rightarrow pqr = 70 \Rightarrow 71$ → Prime
Try $p = 2, q = 3, r = 11 \Rightarrow pqr = 66 + 1 = 67$ → Prime again
Try $p = 3, q = 5, r = 7 \Rightarrow 105 + 1 = 106$ → Composite
Answer: (i) $p = 3$, $q = 5$, $r = 7$ gives $pqr + 1 = 106$ ⇒ composite
(ii) $p = 2$, $q = 3$, $r = 5$ gives $pqr + 1 = 31$ ⇒ prime