Fundamental Theorem of Arithmetic

Step 1: Look at distinct prime factors, not their powers. Step 2: In (168=2^3\times3\times7), the distinct primes are 2, 3, and 7. Step 3: Even if a power is large, count that prime once in the distinct list.

Step 1: Write (125=5\times25). Step 2: (25=5\times5), so (125=5^3). Step 3: 25 is composite, so (25\times5) is not the final prime form.

Step 1: For HCF, take the 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 3 and of (3) is 2, so (2^3\times3^2=72). Step 3: Use highest powers for LCM.

Step 1: Write (216=8\times27). Step 2: (8=2^3) and (27=3^3), so (216=2^3\times3^3). Step 3: (6^3) gives the value, but 6 is not prime.

Step 1: First find (2^4=16). Step 2: (16\times5=80), so the number is 80. Step 3: In a form with powers, it is easier to evaluate the power first.

Step 1: In prime factorisation, a number is written as a product of prime numbers. Step 2: So the final form should contain only prime factors. Step 3: If any composite factor remains, factorise it further.

Step 1: Write (98=2\times49). Step 2: (49=7^2), so (98=2\times7^2). Step 3: 14 is composite, so (7\times14) is not the final prime factorisation.

Step 1: Write (88=8\times11). Step 2: (8=2^3), so (88=2^3\times11). Step 3: The distinct prime factors are 2 and 11.

Step 1: Co-prime numbers have no common factor except 1. Step 2: Therefore, their only common positive factor is 1. Step 3: To identify co-prime numbers, look for common prime factors.

Step 1: (9=3^2) and (16=2^4). Step 2: They have no common prime factor, so they are co-prime. Step 3: If a common prime factor is found, the pair is not co-prime.

Step 1: The common prime factors are 2, 3, and 5. Step 2: The smaller powers are (2^1), (3^1), and (5^1), so the HCF is (2\times3\times5=30). Step 3: Take smaller powers for HCF.

Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^2), (3^2), and (5^1), so (4\times9\times5=180). Step 3: Use highest powers for LCM.

Step 1: Write (252=4\times63). Step 2: (4=2^2) and (63=3^2\times7), so (252=2^2\times3^2\times7). Step 3: Do not keep composite factors in the final answer.

Step 1: Find (3^2=9). Step 2: (9\times7=63), so the number is 63. Step 3: To get the number from prime factorisation, multiply all factors.

Step 1: Write (162=2\times81). Step 2: (81=3^4), so (162=2\times3^4). Step 3: 18 and 9 are composite, so they are not the final prime form.

Step 1: A prime number must have exactly two positive factors. Step 2: 1 has only one positive factor, 1, so it is neither prime nor composite. Step 3: Treating 1 as prime is a common exam mistake.

Step 1: Carefully look at the given form (128=2^7). Step 2: The exponent written on 2 is 7. Step 3: It is important to identify the base and exponent separately.

Step 1: Write (154=2\times77). Step 2: (77=7\times11), so (154=2\times7\times11). Step 3: 14 is composite, so (14\times11) is not the final prime factorisation.

Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (4\times84=336). Step 3: Use this relation directly for two numbers only.