HCF and LCM using prime factorisation

Correct answer: A. (1)

Explanation: Step 1: (25=5^2) and (36=2^2\times3^2). Step 2: They have no common prime factor. Step 3: Therefore, the HCF is (1).

Correct answer: D. (900)

Explanation: Step 1: (25=5^2) and (36=2^2\times3^2). Step 2: There is no common prime factor, so the LCM is their product. Step 3: (25\times36=900), so the answer is (900).

Correct answer: C. (198)

Explanation: Step 1: (11) is prime and (18=2\times3^2). Step 2: They have no common prime factor. Step 3: The LCM of co-prime numbers is their product, so (11\times18=198).

Correct answer: B. (17)

Explanation: Step 1: (34=2\times17) and (51=3\times17). Step 2: The common prime factor is (17). Step 3: Therefore, the HCF is (17).

Correct answer: A. (102)

Explanation: Step 1: (34=2\times17) and (51=3\times17). Step 2: For LCM, take (2), (3), and (17). Step 3: (2\times3\times17=102), so the answer is (102).

Correct answer: B. (12)

Explanation: Step 1: For maximum equal groups, find the HCF. Step 2: (36=2^2\times3^2) and (60=2^2\times3\times5), so HCF (=2^2\times3=12). Step 3: When maximum equal distribution is asked, use HCF.

Correct answer: C. (24)

Explanation: Step 1: For the maximum number of equal piles, find the HCF of (48) and (72). Step 2: (48=2^4\times3) and (72=2^3\times3^2), so HCF (=2^3\times3=24). Step 3: Maximum equal piles are found using HCF.

Correct answer: C. (72)

Explanation: Step 1: The next common ringing time is the LCM of the two intervals. Step 2: (18=2\times3^2) and (24=2^3\times3), so LCM (=2^3\times3^2=72). Step 3: For repeated time questions, use LCM.

Correct answer: D. (90)

Explanation: Step 1: The common sound time is the LCM of (6), (9), and (15). Step 2: (6=2\times3), (9=3^2), and (15=3\times5). Step 3: (2\times3^2\times5=90), so the answer is (90) minutes.

Correct answer: C. (24) cm

Explanation: Step 1: For equal pieces of maximum length, find the HCF. Step 2: (72=2^3\times3^2) and (96=2^5\times3), so HCF (=2^3\times3=24). Step 3: In cutting questions, maximum equal length is found by HCF.

Correct answer: B. (22) m

Explanation: Step 1: For maximum equal length, find the HCF of (66) and (110). Step 2: (66=2\times3\times11) and (110=2\times5\times11). Step 3: The common part is (2\times11=22), so each piece will be (22) m long.

Correct answer: C. (25)

Explanation: Step 1: (125=5^3) and (150=2\times3\times5^2). Step 2: The common prime factor is (5), and the smaller power is (5^2). Step 3: (5^2=25), so the HCF is (25).

Correct answer: B. (750)

Explanation: Step 1: (125=5^3) and (150=2\times3\times5^2). Step 2: The highest powers are (2), (3), and (5^3). Step 3: (2\times3\times125=750), so the LCM is (750).

Correct answer: A. (11)

Explanation: Step 1: (121=11^2) and (143=11\times13). Step 2: The common prime factor is (11). Step 3: Therefore, the HCF is (11).

Correct answer: B. (1573)

Explanation: Step 1: (121=11^2) and (143=11\times13). Step 2: For LCM, take (11^2) and (13). Step 3: (121\times13=1573), so the answer is (1573).

Correct answer: C. (72)

Explanation: Step 1: The common prime factors are (2) and (3). Step 2: The smaller powers are (2^3) and (3^2). Step 3: (8\times9=72), so the HCF is (72).

Correct answer: C. (432)

Explanation: Step 1: The LCM uses the highest powers. Step 2: The highest powers are (2^4) and (3^3). Step 3: (16\times27=432), so the answer is (432).

Correct answer: B. (16)

Explanation: Step 1: (32=2^5), (48=2^4\times3), and (64=2^6). Step 2: The common prime factor is (2), and the smallest power is (2^4). Step 3: (2^4=16), so the HCF is (16).

Correct answer: C. (192)

Explanation: Step 1: (32=2^5), (48=2^4\times3), and (64=2^6). Step 2: The highest powers are (2^6) and (3). Step 3: (64\times3=192), so the LCM is (192).

Correct answer: C. (48)

Explanation: Step 1: (96=2^5\times3) and (144=2^4\times3^2). Step 2: The smaller powers of common factors are (2^4) and (3). Step 3: (16\times3=48), so the HCF is (48).

Correct answer: C. (288)

Explanation: Step 1: (96=2^5\times3) and (144=2^4\times3^2). Step 2: The highest powers are (2^5) and (3^2). Step 3: (32\times9=288), so the LCM is (288).

Correct answer: C. (132)

Explanation: Step 1: First write (2^2=4). Step 2: The number is (4\times3\times11=132). Step 3: Do not ignore powers while reading factorisation.

Correct answer: C. (392)

Explanation: Step 1: (2^3=8) and (7^2=49). Step 2: (8\times49=392). Step 3: First evaluate the powers, then multiply.

Correct answer: B. The larger number

Explanation: Step 1: When the larger number is a multiple of the smaller number, it is divisible by both numbers. Step 2: So the LCM is the larger number itself. Step 3: In such questions, first check whether one number is a multiple of the other.

Correct answer: A. The smaller number

Explanation: Step 1: When one number is a multiple of the other, the smaller number divides both numbers exactly. Step 2: Therefore, the HCF is the smaller number. Step 3: This rule helps solve multiple-based questions quickly.