HCF and LCM using prime factorisation

Step 1: HCF includes only common primes (2) and (3). Step 2: The smaller powers are (2^4) and (3^2), so (M=2^4\times3^2). Step 3: Do not include (5) and (7), as they appear in only one number.

Step 1: Compare the powers of (2) separately. Step 2: (40=2^3\times5), (64=2^6), and (96=2^5\times3), so the power of (2) in the LCM is (6). Step 3: You can compare powers before calculating the full LCM.

Step 1: The smallest common divisible number is the LCM. Step 2: (27=3^3), (45=3^2\times5), and (63=3^2\times7), so LCM (=3^3\times5\times7=945). Step 3: Calculate before choosing, because larger options can mislead.

Step 1: The HCF must divide the LCM. Step 2: (180) is not exactly divisible by (9), so such whole numbers are not possible. Step 3: Check this necessary condition before searching for pairs.

Step 1: When the HCF is (12), (m) and (n) are coprime. Step 2: LCM (=12mn=504), so (mn=42). Step 3: This form makes large-number questions easier.

Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: So product divided by HCF gives the LCM. Step 3: Learn to rearrange the relation, not just memorize it.

Step 1: HCF is (2^2\times3^2=36). Step 2: LCM is (2^2\times3^3\times5=540), so the ratio is (540:36=15:1). Step 3: Always reduce the ratio to its simplest form.

Step 1: The maximum number of rows is found by HCF. Step 2: (132=2^2\times3\times11) and (198=2\times3^2\times11), so HCF (=2\times3\times11=66). Step 3: For maximum equal arrangement, identify HCF.

Step 1: The HCF must be an exact divisor of the LCM. Step 2: (1260) is not exactly divisible by (45), so such two whole numbers are not possible. Step 3: This quick check saves long calculations.

Step 1: A common multiple must be a multiple of the LCM. Step 2: The LCM contains (2^5), (3^4), (5), and (7). Step 3: For a multiple, every required prime power must be present.

Step 1: (42=21\times2) and (231=21\times11). Step 2: Since (2) and (11) are coprime, HCF is (21) and LCM is (21\times2\times11=462). Step 3: Factor out the HCF and check whether the remaining numbers are coprime.

Step 1: The smaller power of (2) in the HCF must be (3). Step 2: The second number has power (5), so (a=3) makes the smaller power (3). Step 3: In power questions, compare smaller and larger values carefully.

Step 1: Compare the powers of (3). Step 2: (72=2^3\times3^2), (90=2\times3^2\times5), and (150=2\times3\times5^2), so the smallest power is (1). Step 3: HCF uses the smallest power.

Step 1: The smallest such number is the LCM of the two given numbers. Step 2: The higher power of (3) is (4), so the smallest number contains (3^4). Step 3: For divisibility, check powers in the LCM.

Step 1: (a=2^2\times5=20). Step 2: Since (ab=420), (b=\frac{420}{20}=21), and (20) and (21) are coprime. Step 3: The coprime condition helps verify the final answer.

Step 1: Product of two numbers equals HCF times LCM. Step 2: The powers of (3) are (1) and (3), so the total power is (4). Step 3: When multiplying powers with the same base, add the exponents.

Step 1: Prime factorise: (48=2^4\times3), (72=2^3\times3^2), and (108=2^2\times3^3). Step 2: The common smallest powers are (2^2) and (3), so HCF is (12). Step 3: For three numbers, take the smallest power across all.

Step 1: (54=2\times3^3), (81=3^4), and (135=3^3\times5). Step 2: Using highest powers, LCM (=2\times3^4\times5=810). Step 3: A prime appearing in only one number still appears in the LCM.

Step 1: Product of two numbers equals HCF times LCM. Step 2: The other number is (\frac{28\times840}{140}=168). Step 3: Simplify the division to reduce mistakes.

Step 1: Product equals HCF (\times) LCM. Step 2: LCM (=\frac{24000}{20}=1200). Step 3: Check the answer using the given relation.

Step 1: (75=3\times5^2) and (125=5^3). Step 2: HCF (=5^2=25) and LCM (=3\times5^3=375), so (L-H=350). Step 3: Find both values separately first.

Step 1: The ratio means dividing LCM by HCF. Step 2: Subtract powers of the same bases: (2^{4-2}\times3^{2-1}\times5=2^2\times3\times5). Step 3: Remember the exponent rule for division.

Step 1: When (r) and (s) are coprime, LCM is (6rs). Step 2: (7) and (8) are coprime, so LCM (=6\times7\times8=336). Step 3: Factor out the HCF and check the remaining numbers.

Step 1: LCM contains the highest power of every prime present. Step 2: (11) appears only in the second number as (11^1), so its power in (L) is (1). Step 3: A prime appearing in only one number is still included in the LCM.

Step 1: Remainder (0) means the number is exactly divisible by all three numbers. Step 2: (36=2^2\times3^2), (54=2\times3^3), and (90=2\times3^2\times5), so LCM (=2^2\times3^3\times5=540). Step 3: The smallest possible value is always the LCM.