Find the HCF of 42, 63, and 140 accurately Use either prime factorisation or the Euclidean algorithm to determine the highest common factor.

Introduction / Context:Finding the HCF (gcd) among three integers can be done sequentially: gcd(a, b, c) = gcd(gcd(a, b), c). This keeps the computation simple and organised.

Given Data / Assumptions:

• a = 42, b = 63, c = 140.
• We want gcd(a, b, c).

Concept / Approach:Either factor each number or apply the Euclidean algorithm pairwise. We will use a quick gcd chain and confirm by factorisation.

Step-by-Step Solution:gcd(42, 63) = 21 (since 42 = 2 * 3 * 7 and 63 = 3^2 * 7).Now gcd(21, 140) = 7 (140 = 2^2 * 5 * 7).Therefore HCF = 7.

Verification / Alternative check:Prime factorisations: 42 = 2 * 3 * 7; 63 = 3^2 * 7; 140 = 2^2 * 5 * 7. The only prime common to all three is 7 with exponent 1.

Why Other Options Are Wrong:21 divides 42 and 63 but not 140; 14 divides 42 and 140 but not 63; 9 does not divide 42 or 140; 28 fails on 63.

Common Pitfalls:Stopping after gcd(42, 63) = 21 without checking against 140; overlooking that a common factor must divide every number.

Final Answer:7