HCF and LCM using prime factorisation

Step 1: In the LCM, take the highest power of (2). Step 2: The powers of (2) are (5), (7), and (4), so the highest power is (7). Step 3: In LCM, do not add powers; take the highest power.

Step 1: Product of two numbers equals HCF times LCM. Step 2: The other number is (\frac{33\times2145}{165}=429). Step 3: Use (165=33\times5) to simplify the division.

Step 1: Remainder (0) means the number is exactly divisible by all three numbers. Step 2: (168=2^3\times3\times7), (210=2\times3\times5\times7), and (280=2^3\times5\times7), so LCM (=2^3\times3\times5\times7=840). Step 3: For the smallest divisible number, find the LCM.

Step 1: HCF uses the smaller power of a common prime. Step 2: The powers of (5) are (3) and (1), so the smaller power is (1). Step 3: Recognise power (1) correctly in the answer.

Step 1: LCM takes the higher power, while HCF takes the lower power. Step 2: The powers of (3) are (2) and (6), so (L) has (6) and (H) has (2). Step 3: Keep the higher-power and lower-power rules separate.

Step 1: (286=26\times11) and (260=26\times10). Step 2: Since (11) and (10) are coprime, HCF is (26) and LCM is (26\times11\times10=2860). Step 3: In options, factor out the HCF and check if the remaining numbers are coprime.

Step 1: For two numbers, HCF (\times) LCM equals the product of the two numbers. Step 2: Therefore, here the product equals (288\times432). Step 3: Apply this relation directly for two numbers.

Step 1: (143=11\times13), (187=11\times17), and (253=11\times23). Step 2: The common prime in all three is (11), so the HCF is (11). Step 3: Identify the prime common to all three numbers.

Step 1: LCM uses the highest power of (3). Step 2: The powers of (3) are (5), (2), and (4), so the highest power is (5). Step 3: Choose the highest power instead of adding powers.

Step 1: After factoring out HCF (81), (r) and (s) are coprime. Step 2: LCM (=81rs=4617), so (rs=57). Step 3: In such questions, first divide the LCM by the HCF.

Step 1: (H=2^4\times3^3\times5) and (L=2^7\times3^6\times5^2). Step 2: (\frac{L}{H}=2^{7-4}\times3^{6-3}\times5^{2-1}=2^3\times3^3\times5). Step 3: In division, subtract powers of the same base.

Step 1: (330=2\times3\times5\times11), (462=2\times3\times7\times11), and (770=2\times5\times7\times11). Step 2: (H=2\times11=22) and (L=2\times3\times5\times7\times11=2310), so (L\div H=105). Step 3: For three numbers, first check common primes and then all distinct primes.

Step 1: The LCM contains all distinct primes appearing in both numbers. Step 2: The distinct primes are (2), (3), (5), and (17), so there are (4). Step 3: Count only distinct prime bases, not their powers.

Step 1: The HCF must exactly divide the LCM. Step 2: (1248\div96=13), which is a whole number, so such whole numbers can exist. Step 3: For existence checks, first test this necessary condition.

Step 1: HCF uses only common prime factors. Step 2: The common primes are (2), (3), and (5), with smaller powers (2^4), (3^2), and (5). Step 3: Choose the smaller power for each prime base separately.

Step 1: LCM takes the highest power of every prime present. Step 2: The highest powers are (2^5), (3^5), (5^2), (7), and (13). Step 3: A prime appearing in only one number must also be included.

Step 1: Product of two numbers equals HCF times LCM. Step 2: The other number is (\frac{45\times3465}{315}=495). Step 3: Notice (315=45\times7) to calculate quickly.

Step 1: HCF is (2^4\times3^2\times5), and LCM is (2^7\times3^6\times5^3\times7\times11). Step 2: On division, subtract powers to get (2^3\times3^4\times5^2\times7\times11). Step 3: In ratios, subtract exponents of the same base.

Step 1: Let the numbers be (30m) and (30n), where (m) and (n) are coprime. Step 2: (30mn=2730), so (mn=91=7\times13); the unordered coprime pairs are ((1,91)) and ((7,13)). Step 3: Do not count reversed order as a new pair.

Step 1: In each case, divisor minus remainder is (7), so adding (7) to the number makes it divisible by all three divisors. Step 2: The LCM of (45), (75), and (90) is (450), so the number is (450-7=443). Step 3: Spot the common difference and subtract it from the LCM.