HCF and LCM using prime factorisation

Step 1: After taking out HCF (18), (r) and (s) are coprime. Step 2: LCM (=18rs=990), so (rs=55). Step 3: Divide by the HCF to simplify such questions.

Step 1: (132=2^2\times3\times11) and (308=2^2\times7\times11). Step 2: The common smaller powers are (2^2) and (11), so HCF (=44). Step 3: Compare prime factors before choosing the statement.

Step 1: LCM takes the higher power of a common prime. Step 2: The powers of (2) are (6) and (4), so the higher power is (6). Step 3: In LCM, powers are not added; only the higher power is taken.

Step 1: Check prime factors: (66=2\times3\times11), (88=2^3\times11), and (121=11^2). Step 2: The distinct primes in the LCM are (2), (3), and (11), so the count is (3). Step 3: Do not count powers as separate primes.

Step 1: The HCF must exactly divide the LCM. Step 2: (1260) is not exactly divisible by (45), so such whole numbers are not possible. Step 3: Check this necessary condition before searching for pairs.

Step 1: For maximum equal length, find the HCF. Step 2: (80=2^4\times5) and (128=2^7), so HCF (=2^4=16). Step 3: For maximum equal cutting, use HCF.

Step 1: The smallest number divisible by all is the LCM. Step 2: (36=2^2\times3^2), (100=2^2\times5^2), and (150=2\times3\times5^2), so LCM (=2^2\times3^2\times5^2=900). Step 3: Always verify the final multiplication before choosing an option.

Step 1: Such a smallest number is the LCM of the three numbers. Step 2: (36=2^2\times3^2), (100=2^2\times5^2), and (150=2\times3\times5^2), so LCM (=2^2\times3^2\times5^2=900). Step 3: Take the highest power of each prime.

Step 1: In (\frac{L}{H}), divide the LCM by the HCF. Step 2: Subtract powers of the same bases: (2^{6-2}\times3^{3-1}\times5^{2-1}\times7=2^4\times3^2\times5\times7). Step 3: Use exponent subtraction in division.

Step 1: Product of two numbers equals HCF times LCM. Step 2: LCM (=\frac{286\times429}{143}=858). Step 3: Divide (286) by (143) first to get (2), then multiply (2\times429).

Step 1: (q=p\times7), so (p) exactly divides (q). Step 2: When one number exactly divides the other, the smaller number is the HCF. Step 3: First check whether one number is a multiple of the other.

Step 1: The maximum number of equal parts is found using HCF. Step 2: (96=2^5\times3), (144=2^4\times3^2), and (192=2^6\times3), so HCF (=2^4\times3=48). Step 3: For maximum equal division, use HCF.

Step 1: HCF includes only primes common to all three numbers. Step 2: (2) and (3) are common, but (5) is not in the third number; the smallest powers are (2^2) and (3^2). Step 3: Do not include a prime that is not present in every number.

Step 1: (44=2^2\times11), (77=7\times11), and (121=11^2). Step 2: The highest power of (11) in the LCM is (2). Step 3: For LCM, choose the highest power.

Step 1: When (u) and (v) are coprime, the LCM of (30u) and (30v) is (30uv). Step 2: Since (uv=26), LCM (=30\times26=780). Step 3: Factoring out the HCF simplifies the question.

Step 1: LCM (=\frac{216\times360}{72}=1080). Step 2: Then (\frac{1080}{72}=15), so the value of the ratio is (15). Step 3: First find the LCM, then simplify the ratio.

Step 1: A common multiple must contain all prime powers required by both numbers. Step 2: The highest powers are (2^6), (3^3), (5), and (7). Step 3: Do not miss any required prime factor while checking a multiple.

Step 1: HCF contains only primes present in both numbers. Step 2: (2), (3), and (7) are common, but (5) appears only in the second number. Step 3: First identify common primes, then choose their smaller powers.

Step 1: LCM takes the higher power of (5), and HCF takes the lower power. Step 2: The powers of (5) are (1) and (3), so (L) has power (3) and (H) has power (1). Step 3: Do not interchange the two rules.

Step 1: Product of two numbers equals HCF times LCM. Step 2: (11) appears only in the LCM as (11^1), so its power in the product is (1). Step 3: In multiplication, add powers of the same prime.