Let p, q and r be three distinct prime numbers. Check whether $p \cdot q \cdot r + q$ is a composite number or not. Further, give an example for 3 distinct primes p, q, r such that
(i) $p \cdot q \cdot r + 1$ is a composite number.
(ii) $p \cdot q \cdot r + 1$ is a prime number.
Generated by claude-sonnet-4-6 · 2026-06-15 10:31 · grounding rag
Model Answer
Checking p·q·r + q:
$p \cdot q \cdot r + q = q(pr + 1)$
Since q > 1 and (pr + 1) > 1, this expression has factors other than 1 and itself. Hence, p·q·r + q is always a composite number.
(i) p·q·r + 1 is composite:
Let p = 2, q = 3, r = 5.
$2 × 3 × 5 + 1 = 31$ — this is prime. Try p = 2, q = 7, r = 11:
$2 × 7 × 11 + 1 = 155 = 5 × 31$ → composite. ✓
(So p = 2, q = 7, r = 11 works.)
(ii) p·q·r + 1 is prime:
Let p = 2, q = 3, r = 5.
$2 × 3 × 5 + 1 = 31$, which is prime. ✓
Source: Chapter 1, Section 1.2 – The Fundamental Theorem of Arithmetic
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Explanation
- The key step for the first part is factoring out q: $q(pr+1)$ immediately shows two factors both greater than 1, proving it composite. Examiners want this algebraic step shown clearly.
- For part (i) and (ii), you just need one valid example each with verification. The examiner checks that you compute the product, add 1, and confirm composite/prime. Don't overcomplicate — one correct example with the calculation shown is full marks.
- Note: the same set of primes (2, 3, 5) gives a prime in (ii), so use a different set for (i).