Compute the HCF using prime factorizations quickly Find the highest common factor (HCF) of 1485 and 4356.

Introduction / Context:HCF (gcd) between two integers is easiest via either the Euclidean algorithm or prime factorisation. Recognising square structures can speed factorisation substantially.

Given Data / Assumptions:

• Numbers: 1485 and 4356.
• We aim to find gcd(1485, 4356).

Concept / Approach:Factor both: 1485 ends with 5, so divisible by 5; 4356 is 4 * 1089 and 1089 is 33^2, which is 3^2 * 11^2. Take minimum exponents for common primes.

Step-by-Step Solution:1485 = 5 * 297 = 5 * (27 * 11) = 3^3 * 5 * 11.4356 = 4 * 1089 = 2^2 * (33^2) = 2^2 * 3^2 * 11^2.Common primes: 3 and 11. Minimum exponents: 3^2 and 11^1.Therefore HCF = 3^2 * 11 = 9 * 11 = 99.

Verification / Alternative check:Divide: 1485 ÷ 99 = 15; 4356 ÷ 99 = 44. Both are integers, confirming gcd = 99.

Why Other Options Are Wrong:189 = 9 * 21 includes factor 7 (not common); 89 and 83 are primes that do not divide 1485; 27 divides 1485 but not 4356 exactly.

Common Pitfalls:Stopping after finding a small common factor like 3 or 11; forgetting to include both common primes at their minimum exponents.

Final Answer:99