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

Problem:
Given three prime numbers \( p, q, r \), determine if \( pqr + 1 \) and \( pqr + q \) are prime or composite.

We will test different prime number combinations.

Step 1: Test with small primes
Let \( p = 2, q = 3, r = 5 \)
Then:\[pqr = 2 \cdot 3 \cdot 5 = 30\]
(i) Evaluate \( pqr + q = 30 + 3 = 33 \)
33 is divisible by 3 and 11 ⇒ It is composite

(ii) Evaluate \( pqr + 1 = 30 + 1 = 31 \)
31 is a prime number

Step 2: Test with \( p = 2, q = 3, r = 7 \)
\[pqr = 2 \cdot 3 \cdot 7 = 42\Rightarrow pqr + 1 = 43\]
43 is a prime number

Step 3: Test with \( p = 2, q = 5, r = 7 \)
\[pqr = 2 \cdot 5 \cdot 7 = 70 \Rightarrow pqr + 1 = 71\]
71 is a prime number

Step 4: Test with \( p = 2, q = 3, r = 11 \)
\[pqr = 2 \cdot 3 \cdot 11 = 66 \Rightarrow pqr + 1 = 67\]
67 is a prime number

Step 5: Test with \( p = 3, q = 5, r = 7 \)
\[pqr = 3 \cdot 5 \cdot 7 = 105 \Rightarrow pqr + 1 = 106\]
106 is divisible by 2 ⇒ It is composite

Conclusions:
(i) \( \boxed{p = 3, q = 5, r = 7} \Rightarrow pqr + 1 = 106 \) → composite
(ii) \( \boxed{p = 2, q = 3, r = 5} \Rightarrow pqr + 1 = 31 \) → prime