HCF and LCM using prime factorisation

Correct answer: B. (2^3\times3^2)

Explanation: Step 1: HCF uses only the common prime factors. Step 2: The common primes are (2) and (3), and the smaller powers are (2^3) and (3^2). Step 3: For HCF, always choose the smaller power of each common prime.

Correct answer: A. (2^3\times3^2\times5\times7)

Explanation: Step 1: LCM takes the highest power of every prime factor present. Step 2: The required powers are (2^3), (3^2), (5), and (7). Step 3: Include primes that appear in even one of the numbers.

Correct answer: A. (180)

Explanation: Step 1: For two numbers, product of numbers equals HCF times LCM. Step 2: The other number is (\frac{24\times720}{96}=180). Step 3: Write the relation first and then divide carefully.

Correct answer: C. (480)

Explanation: Step 1: The smallest number divisible by all given numbers is their LCM. Step 2: (32=2^5), (48=2^4\times3), and (80=2^4\times5), so LCM (=2^5\times3\times5=480). Step 3: Do not forget the highest power of each prime.

Correct answer: B. (72)

Explanation: Step 1: The greatest number dividing all given numbers is the HCF. Step 2: (144=2^4\times3^2), (216=2^3\times3^3), and (360=2^3\times3^2\times5), so HCF (=2^3\times3^2=72). Step 3: Use only primes common to all numbers.

Correct answer: A. Such numbers are not possible

Explanation: Step 1: The HCF must exactly divide the LCM. Step 2: Since (378\div18=21), such numbers are possible, so the statement saying impossible is not correct. Step 3: Always check divisibility before deciding validity.

Correct answer: A. (2^2\times3^2\times5\times11)

Explanation: Step 1: HCF is (2^2\times3\times5), and LCM is (2^4\times3^3\times5^2\times11). Step 2: On division, subtract powers, giving (2^2\times3^2\times5\times11). Step 3: Prime factor form makes ratio questions faster.

Correct answer: A. (140)

Explanation: Step 1: The time after which all ring together is the LCM of the intervals. Step 2: (20=2^2\times5), (28=2^2\times7), and (35=5\times7), so LCM (=2^2\times5\times7=140). Step 3: For repeated time events, use LCM.

Correct answer: B. (42)

Explanation: Step 1: The maximum number of identical packets is found using HCF. Step 2: (84=2^2\times3\times7) and (126=2\times3^2\times7), so HCF (=2\times3\times7=42). Step 3: For maximum equal distribution, use HCF.

Correct answer: C. (365)

Explanation: Step 1: If the remainder is (5), then subtracting (5) makes the number divisible by (18), (24), and (30). Step 2: Their LCM is (360), so the smallest number is (360+5=365). Step 3: For common remainder questions, subtract the remainder first.

Correct answer: B. (a=3), (b=2)

Explanation: Step 1: HCF uses the smaller power of each common prime. Step 2: For (2), the smaller power must be (3), so (a=3) is possible; for (3), (b=2) is possible. Step 3: Check each unknown power separately.

Correct answer: A. (a=4), (b=2)

Explanation: Step 1: LCM uses the highest power of each prime. Step 2: The highest power of (3) must be (4), so (a=4); the highest power of (5) must be (2), so (b=2). Step 3: In LCM power questions, identify the maximum power.

Correct answer: A. Product of (a) and (b) is (30) and they are coprime

Explanation: Step 1: If HCF is (30), the numbers can be written as (30a) and (30b), where (a) and (b) are coprime. Step 2: LCM becomes (30ab=900), so (ab=30). Step 3: Factor out the HCF to simplify such problems.

Correct answer: A. (54) and (180)

Explanation: Step 1: A correct pair must satisfy both HCF and LCM. Step 2: (54=18\times3) and (180=18\times10), where (3) and (10) are coprime, so HCF is (18) and LCM is (18\times3\times10=540). Step 3: In options, check HCF first.

Correct answer: D. (221)

Explanation: Step 1: Coprime numbers have HCF (1). Step 2: Product equals HCF times LCM, so product (=1\times221=221). Step 3: For coprime numbers, LCM equals the product.

Correct answer: A. (630)

Explanation: Step 1: For two numbers, product equals HCF times LCM. Step 2: LCM (=\frac{126\times315}{63}=630). Step 3: Simplify before multiplying large numbers.

Correct answer: C. (5)

Explanation: Step 1: LCM includes all distinct prime factors appearing in the numbers. Step 2: The distinct primes are (2), (3), (5), (7), and (11), so there are (5). Step 3: Do not count powers as separate primes.

Correct answer: A. (403)

Explanation: Step 1: Subtracting (3) makes the number divisible by (16), (20), and (25). Step 2: (16=2^4), (20=2^2\times5), and (25=5^2), so LCM (=2^4\times5^2=400). Hence the number is (400+3=403). Step 3: Add the common remainder at the end.

Correct answer: B. (2^2\times3)

Explanation: Step 1: HCF of three numbers includes primes common to all three. Step 2: (2) and (3) are common, with smallest powers (2^2) and (3). Step 3: For three numbers, compare all three powers before choosing.

Correct answer: A. (10240)

Explanation: Step 1: Product of two numbers equals HCF (\times) LCM. Step 2: (16\times640=10240), so the product is (10240). Step 3: Apply this relation directly only for two numbers.

Correct answer: B. (1)

Explanation: Step 1: HCF uses the smaller power of a common prime. Step 2: The powers of (5) are (1) and (2), so the smaller power is (1). Step 3: A power of (1) is often not written, but it still matters.

Correct answer: A. Their HCF is (30)

Explanation: Step 1: (150=2\times3\times5^2) and (210=2\times3\times5\times7). Step 2: The common smaller powers are (2), (3), and (5), so HCF (=30). Step 3: For statement-based questions, prime factorise first.

Correct answer: B. (2^3\times3^2\times5\times7)

Explanation: Step 1: LCM takes the highest power of every prime. Step 2: Combining (2^3), (3^2), (5), and (7) gives (2^3\times3^2\times5\times7). Step 3: Include primes that occur in only one number too.

Correct answer: C. (36)

Explanation: Step 1: Product of two numbers equals HCF times LCM. Step 2: HCF (=\frac{15120}{420}=36). Step 3: Simplify the division to save time.

Correct answer: B. (32)

Explanation: Step 1: For maximum equal length, find the HCF. Step 2: (96=2^5\times3) and (160=2^5\times5), so HCF (=2^5=32). Step 3: For maximum equal cutting or sharing, use HCF.