Fundamental Theorem of Arithmetic

Step 1: Write (108=4\times27). Step 2: (4=2^2) and (27=3^3), so (108=2^2\times3^3). Step 3: 12 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 mistake.

Step 1: Look at the given form (96=2^5\times3). Step 2: The exponent written on 2 is 5. Step 3: While reading powers, identify the base and exponent separately.

Step 1: Write (126=2\times63). Step 2: (63=9\times7=3^2\times7), so (126=2\times3^2\times7). Step 3: 6 and 21 are composite, so they are not final answers.

Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (5\times60=300). Step 3: Use this formula when the question is about two numbers.

Step 1: Write 81 as (9\times9). Step 2: Each 9 is (3^2), so (81=3^4). Step 3: 9 is composite, so (9^2) is not prime factorisation.

Step 1: Write (150=15\times10). Step 2: (15=3\times5) and (10=2\times5), so the distinct prime factors are 2, 3, and 5. Step 3: Write a repeated factor only once in a distinct list.

Step 1: The common prime factors are 2 and 3. Step 2: The smaller powers are (2^1) and (3^1), so the HCF is (2\times3=6). Step 3: Take smaller powers for HCF.

Step 1: For LCM, take higher powers. Step 2: The higher power of (2) is 3 and of (3) is 2, so (2^3\times3^2=72). Step 3: LCM includes the highest powers of all prime factors.

Step 1: Write (200=8\times25). Step 2: (8=2^3) and (25=5^2), so (200=2^3\times5^2). Step 3: 20 and 10 are composite, so they are not prime factorisation.

Step 1: Multiply the given prime factors. Step 2: (2\times3\times7=42), so the number is 42. Step 3: To get the number from factorisation, multiply all factors.

Step 1: Find (2^4=16). Step 2: (16\times3=48), so the number is 48. Step 3: First evaluate the power, then multiply.

Step 1: In prime factorisation, all factors must be prime. Step 2: In (4\times3), 4 is not prime, so this is not prime factorisation. Step 3: If a composite factor appears, factorise it further.

Step 1: Write (132=12\times11). Step 2: (12=2^2\times3) and 11 is prime, so (132=2^2\times3\times11). Step 3: Do not leave 12 in the final form because it is composite.

Step 1: (14=2\times7) and (25=5^2). Step 2: They have no common prime factor, so they are co-prime. Step 3: Co-prime numbers have HCF 1.

Step 1: Write (225=15\times15). Step 2: Each 15 is (3\times5), so (225=3^2\times5^2). Step 3: 15 is composite, so (15\times15) is not the final prime form.

Step 1: The common prime factors are 2 and 5. Step 2: The smaller powers are (2^2) and (5^1), so HCF is (4\times5=20). Step 3: For HCF, take smaller powers of common factors only.

Step 1: For LCM, take the highest powers of all prime factors. Step 2: (2^3\times3\times5=8\times3\times5=120). Step 3: In LCM, include prime factors that appear in either number.

Step 1: A number divisible by both 2 and 3 is also divisible by 6. Step 2: 36 is even and its digit sum is 9, so it is also divisible by 3. Step 3: Divisibility checks help in prime factorisation.

Step 1: Find (2^2=4) and (5^2=25). Step 2: (4\times3\times25=300). Step 3: In questions with powers, simplify the powers first.

Step 1: This theorem is about writing positive integers greater than 1 as products of primes. Step 2: So it discusses numbers greater than 1. Step 3: Do not include 1 in the usual prime factorisation statement.

Step 1: Write (210=21\times10). Step 2: (21=3\times7) and (10=2\times5), so (210=2\times3\times5\times7). Step 3: In the final form, all factors must be prime.

Step 1: Multiply the given prime factors. Step 2: (3\times5\times7=105), so the number is 105. Step 3: To get the original number from prime factorisation, multiply all factors.

Step 1: Write (180=18\times10). Step 2: (18=2\times3^2) and (10=2\times5), so (180=2^2\times3^2\times5). Step 3: Combine repeated prime factors using powers.

Step 1: Co-prime numbers have HCF 1. Step 2: For two numbers, product (=) HCF (\times) LCM. Step 3: Therefore, the LCM of co-prime numbers is equal to their product.