Fundamental Theorem of Arithmetic

Step 1: (25=5^2) and (36=2^2\times3^2). Step 2: They have no common prime factor, so they are co-prime. Step 3: If a common prime factor appears, the pair is not co-prime.

Step 1: The common prime factors are 2, 3, and 7. Step 2: The smaller powers are (2^2), (3^1), and (7^1), so (4\times3\times7=84). Step 3: HCF uses smaller powers.

Step 1: For LCM, take the highest powers of all prime factors. Step 2: (2^3\times3^2\times7=8\times9\times7=504). Step 3: LCM uses highest powers.

Step 1: Write (360=36\times10). Step 2: (36=2^2\times3^2) and (10=2\times5), so (360=2^3\times3^2\times5). Step 3: Combine repeated prime factors using powers.

Step 1: Find (2^2=4). Step 2: (4\times3\times5=60), so the number is 60. Step 3: To get the number from prime factorisation, multiply all factors.

Step 1: Divide 243 repeatedly by 3. Step 2: (243=3^5), so this is the prime factorisation. Step 3: 9 and 81 are composite, so do not keep them in the final prime form.

Step 1: A prime number has exactly two positive factors. Step 2: 1 has only one positive factor, 1. Step 3: Therefore, 1 is neither prime nor composite.

Step 1: Look at the given form (625=5^4). Step 2: The exponent written on 5 is 4. Step 3: It is important to identify the base and the exponent separately.

Step 1: Write (182=2\times91). Step 2: (91=7\times13), so (182=2\times7\times13). Step 3: 14 is composite, so (14\times13) is not the final prime factorisation.

Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (6\times72=432). Step 3: Apply this relation directly for two numbers.

Step 1: 289 is a perfect square. Step 2: (289=17\times17=17^2). Step 3: 1 is not treated as a separate factor in prime factorisation.

Step 1: Write (231=21\times11). Step 2: (21=3\times7), so (231=3\times7\times11). Step 3: The distinct prime factors are 3, 7, and 11.

Step 1: The common prime factors are 2 and 3. Step 2: The smaller powers are (2^1) and (3^2), so the HCF is (2\times9=18). Step 3: Take smaller powers for HCF.

Step 1: For LCM, take the highest powers. Step 2: (2^2\times3^3=4\times27=108). Step 3: LCM includes the highest powers of all prime factors.

Step 1: Write (375=3\times125). Step 2: (125=5^3), so (375=3\times5^3). Step 3: 15 and 25 are composite, so do not keep them in the final prime form.

Step 1: Find (3^2=9). Step 2: (2\times9\times7=126), so the number is 126. Step 3: To get the number from prime factorisation, simplify powers first.

Step 1: Find (2^6=64). Step 2: (64\times3=192), so the number is 192. Step 3: First evaluate the power, then multiply.

Step 1: In prime factorisation, every factor must be prime. Step 2: In (9\times5), 9 is not prime. Step 3: If a composite factor appears, break it into prime factors.

Step 1: Write (208=16\times13). Step 2: (16=2^4) and 13 is prime, so (208=2^4\times13). Step 3: Since 16 is composite, write it as (2^4) in the final form.

Step 1: (26=2\times13) and (45=3^2\times5). Step 2: They have no common prime factor, so they are co-prime. Step 3: Co-prime numbers have HCF 1.