HCF and LCM using prime factorisation

Correct answer: A. (1)

Explanation: Step 1: (14=2\times7) and (25=5^2). Step 2: They have no common prime factor. Step 3: Therefore, the HCF is (1), and such numbers are co-prime.

Correct answer: C. (350)

Explanation: Step 1: (14=2\times7) and (25=5^2). Step 2: Since they have no common prime factor, include all factors for LCM. Step 3: (2\times7\times25=350), so the LCM is (350).

Correct answer: D. (120)

Explanation: Step 1: (8=2^3) and (15=3\times5). Step 2: They have no common prime factor. Step 3: The LCM of co-prime numbers is their product, so (8\times15=120).

Correct answer: C. (7)

Explanation: Step 1: (21=3\times7) and (35=5\times7). Step 2: The common prime factor is (7). Step 3: Therefore, the HCF is (7).

Correct answer: C. (105)

Explanation: Step 1: (21=3\times7) and (35=5\times7). Step 2: For LCM, take (3), (5), and (7). Step 3: (3\times5\times7=105), so the LCM is (105).

Correct answer: B. (12)

Explanation: Step 1: For the maximum number of equal packets, find the HCF. Step 2: (24=2^3\times3) and (36=2^2\times3^2), so HCF (=2^2\times3=12). Step 3: When maximum equal groups are asked, use HCF.

Correct answer: C. (15)

Explanation: Step 1: For maximum equal boxes, find the HCF of (30), (45), and (60). Step 2: (30=2\times3\times5), (45=3^2\times5), (60=2^2\times3\times5). Step 3: The common part is (3\times5=15), so (15) boxes can be made.

Correct answer: C. (36)

Explanation: Step 1: The time when both bells ring together is the LCM of the two times. Step 2: (12=2^2\times3) and (18=2\times3^2), so LCM (=2^2\times3^2=36). Step 3: For repeated time events, use LCM.

Correct answer: B. (60)

Explanation: Step 1: The next common flashing time is the LCM of (10), (15), and (20). Step 2: (10=2\times5), (15=3\times5), (20=2^2\times5). Step 3: (2^2\times3\times5=60), so the answer is (60) seconds.

Correct answer: C. (24) cm

Explanation: Step 1: For equal pieces of maximum length, find the HCF. Step 2: (48=2^4\times3) and (72=2^3\times3^2), so HCF (=2^3\times3=24). Step 3: For cutting or dividing into maximum equal parts, use HCF.

Correct answer: B. (4) m

Explanation: Step 1: For maximum equal length, find the HCF of (20) and (28). Step 2: (20=2^2\times5) and (28=2^2\times7). Step 3: The common part is (2^2=4), so each piece will be (4) m long.

Correct answer: C. (27)

Explanation: Step 1: (54=2\times3^3) and (81=3^4). Step 2: The common prime factor is (3), and the smaller power is (3^3). Step 3: (3^3=27), so the HCF is (27).

Correct answer: A. (162)

Explanation: Step 1: (54=2\times3^3) and (81=3^4). Step 2: Take the highest powers (2) and (3^4). Step 3: (2\times81=162), so the LCM is (162).

Correct answer: C. (21)

Explanation: Step 1: (63=3^2\times7) and (84=2^2\times3\times7). Step 2: The common factors are (3) and (7). Step 3: (3\times7=21), so the HCF is (21).

Correct answer: C. (252)

Explanation: Step 1: (63=3^2\times7) and (84=2^2\times3\times7). Step 2: The highest powers are (2^2), (3^2), and (7). Step 3: (4\times9\times7=252), so the LCM is (252).

Correct answer: B. (24)

Explanation: Step 1: The common prime factors are (2) and (3). Step 2: The smaller powers are (2^3) and (3). Step 3: (2^3\times3=24), so the HCF is (24).

Correct answer: C. (480)

Explanation: Step 1: The LCM contains the highest powers of all prime factors. Step 2: The highest powers are (2^5), (3), and (5). Step 3: (32\times3\times5=480), so the answer is (480).

Correct answer: A. (5)

Explanation: Step 1: (25=5^2), (40=2^3\times5), and (50=2\times5^2). Step 2: The common prime factor in all three numbers is (5). Step 3: The smallest power is (5), so the HCF is (5).

Correct answer: B. (200)

Explanation: Step 1: (25=5^2), (40=2^3\times5), and (50=2\times5^2). Step 2: The highest powers are (2^3) and (5^2). Step 3: (8\times25=200), so the LCM is (200).

Correct answer: D. (44)

Explanation: Step 1: (88=2^3\times11) and (132=2^2\times3\times11). Step 2: The smaller powers of common factors are (2^2) and (11). Step 3: (4\times11=44), so the HCF is (44).

Correct answer: B. (264)

Explanation: Step 1: (88=2^3\times11) and (132=2^2\times3\times11). Step 2: The highest powers are (2^3), (3), and (11). Step 3: (8\times3\times11=264), so the LCM is (264).

Correct answer: C. (84)

Explanation: Step 1: First simplify the given prime powers. Step 2: Since (2^2=4), the number is (4\times3\times7=84). Step 3: Reading prime factorisation correctly helps in HCF and LCM questions.

Correct answer: B. (200)

Explanation: Step 1: (2^3=8) and (5^2=25). Step 2: Multiplying them gives (8\times25=200). Step 3: Always simplify powers first and then multiply.

Correct answer: A. They are co-prime

Explanation: Step 1: If the HCF of two numbers is (1), they have no common factor except (1). Step 2: Such numbers are called co-prime. Step 3: Co-prime numbers need not both be prime numbers.

Correct answer: A. Smaller power of common factors

Explanation: Step 1: HCF is made only from common prime factors. Step 2: Among common factors, the smaller power is chosen because it divides both numbers. Step 3: Remember the rule: common factors with smaller powers for HCF.