Fundamental Theorem of Arithmetic

Step 1: This theorem is connected with prime factorisation. Step 2: Every positive integer greater than 1 can be written as a product of prime numbers. Step 3: In exams, remember it through prime factorisation.

Step 1: Write (54=2\times27). Step 2: (27=3^3), so (54=2\times3^3). Step 3: 6 and 9 are composite, so do not leave them in the final prime form.

Step 1: (80) can be written as (16\times5). Step 2: (16=2^4), so (80=2^4\times5). Step 3: In the final prime factorisation, keep only prime numbers.

Step 1: Changing the order in multiplication does not change the product. Step 2: (2\times5\times3) and (3\times2\times5) both give 30. Step 3: In prime factorisation, the set of factors matters, not the order.

Step 1: Write (112=16\times7). Step 2: (16=2^4), so (112=2^4\times7). Step 3: Count repeated prime factors to write the exponent.

Step 1: First find (2^3=8). Step 2: (8\times3\times7=168), so the number is 168. Step 3: In expressions with powers, evaluate the power first.

Step 1: Write (99=9\times11). Step 2: (9=3^2) and 11 is prime, so (99=3^2\times11). Step 3: 9 and 33 are composite, so do not keep them in the final answer.

Step 1: A prime number has exactly two positive factors. Step 2: 47 is divisible only by 1 and 47, while 51, 57, and 63 are composite. Step 3: Check small numbers by 2, 3, 5, and 7.

Step 1: A composite number has more than two positive factors. Step 2: (91=7\times13), so it is composite. Step 3: If a number can be written as a product of smaller primes, it is composite.

Step 1: Write (105=15\times7). Step 2: (15=3\times5), so (105=3\times5\times7). Step 3: 15 is composite, so do not keep it in the final prime factorisation.

Step 1: Look at distinct prime factors, not their powers. Step 2: In (180=2^2\times3^2\times5), the distinct primes are 2, 3, and 5. Step 3: Even with a higher power, count the same prime once.

Step 1: 169 is a perfect square. Step 2: (169=13\times13=13^2). Step 3: In prime factorisation, 1 is not written as a separate factor.

Step 1: For HCF, take smaller powers of common prime factors. Step 2: The smaller power of (2) is 2 and of (3) is 1, so (2^2\times3=12). Step 3: Use smaller powers for HCF.

Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest power of (2) is 4 and of (3) is 2, so (2^4\times3^2=144). Step 3: Use highest powers for LCM.

Step 1: Write (288=32\times9). Step 2: (32=2^5) and (9=3^2), so (288=2^5\times3^2). Step 3: 32 and 9 are composite, so write prime powers in the final form.

Step 1: First find (3^3=27). Step 2: (27\times5=135), so the number is 135. Step 3: In a form with powers, evaluate the power first.

Step 1: In prime factorisation, only prime factors are written. Step 2: 1 is not prime because it does not have exactly two positive factors. Step 3: Therefore, (1\times n) is not treated as prime factorisation.

Step 1: Write (147=3\times49). Step 2: (49=7^2), so (147=3\times7^2). Step 3: 21 and 49 are composite, so do not keep them in the final form.

Step 1: Write (124=4\times31). Step 2: (4=2^2) and 31 is prime, so (124=2^2\times31). Step 3: The distinct prime factors are 2 and 31.

Step 1: Co-prime numbers have no common factor except 1. Step 2: Therefore, their HCF is 1. Step 3: To identify co-prime numbers, check common prime factors.

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.