HCF and LCM using prime factorisation

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).

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).

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).

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).

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).

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).

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).

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.

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

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.

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.

Step 1: (2160=2^4\times 3^3\times 5) and (3780=2^2\times 3^3\times 5\times 7). Step 2: For HCF, take the smaller exponents of common prime factors, so (2^2\times 3^3\times 5=540). Step 3: In such questions, carefully choose the smaller powers.

Step 1: For LCM, take the greater exponent of every prime factor present. Step 2: This gives (2^3\times 3^2\times 5\times 7=2520). Step 3: While finding LCM, do not leave out any prime factor.

Step 1: Since the product is directly given, the other number is (15120\div 216). Step 2: (15120\div 216=70). On checking, the HCF of (216) and (70) is (2), so the given HCF (36) is inconsistent. Step 3: In exams, if conditions conflict, first compute the basic quotient and then verify the condition.

Step 1: Divide the total product by the given number to get the other number. Step 2: (15120\div 216=70), so the other number is (70). Step 3: In such questions, find the quotient accurately before doing any extra work.

Step 1: For two numbers, product of the numbers equals HCF times LCM. Step 2: The other number (=\frac{18\times 540}{90}=108). Step 3: Apply this relation directly only for two numbers.

Step 1: For two numbers, HCF times LCM equals the product of the two numbers. Step 2: Adding exponents gives (2^{3+2}\times 3^{2+4}\times 5\times 7=2^5\times 3^6\times 5\times 7). Step 3: When the product is asked, you need not find HCF and LCM separately.

Step 1: HCF contains only the prime factors common to all three numbers. Step 2: The smallest exponent of (2) is (3) and of (3) is (1). (5) and (7) are not common to all. Hence the answer is (2^3\times 3). Step 3: For three numbers, first identify prime factors common to all.

Step 1: LCM takes the greatest exponent of every prime factor present. Step 2: The greatest exponent of (2) is (4), of (3) is (5), and of (5) is (2). So the answer is (2^4\times 3^5\times 5^2). Step 3: Any prime appearing in at least one number must appear in the LCM.

Step 1: For two numbers, their product equals HCF times LCM. Step 2: The exponent of (2) in the product is (2+5=7). Step 3: If only one prime exponent is asked, add only that prime's exponents.