State true or false for each of the following statements and justify in each case :
Generated by claude-sonnet-4-6 · 2026-06-15 10:31 · grounding rag
Model Answer
(i) True.
$2 \times 3 \times 5 \times 7 + 7 = 210 + 7 = 217$
Taking 7 as common: $= 7(2 \times 3 \times 5 + 1) = 7 \times 31$
Since 217 has factors other than 1 and itself (7 and 31), it is a composite number.
(ii) True.
$2 \times 3 \times 5 \times 7 + 1 = 210 + 1 = 211$
Wait — 211 is actually a prime number (not divisible by 2, 3, 5, 7, 11, 13; and $\sqrt{211} < 15$).
Correction: (ii) is FALSE. $2 \times 3 \times 5 \times 7 + 1 = 211$, which is a prime number, not composite.
Source: Chapter 1, Section 1.2 – The Fundamental Theorem of Arithmetic
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Explanation
- For (i), factoring out 7 immediately shows it is composite — this is the key step examiners look for.
- For (ii), many students carelessly mark it True assuming any such expression must be composite. You must actually compute: 211 is prime (check divisibility by all primes up to $\sqrt{211} \approx 14.5$, i.e., 2, 3, 5, 7, 11, 13 — none divide 211). The answer is False.
- Always compute the value and verify; do not assume.