Fundamental Theorem of Arithmetic

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.

Step 1: This theorem is related to writing numbers as products of prime factors. Step 2: Every positive integer greater than 1 can be written as a product of primes. Step 3: In this chapter, connect it with factorisation and HCF.

Step 1: 48 can be written as (16\times3). Step 2: (16=2^4), so (48=2^4\times3). Step 3: Do not leave composite factors like 4 or 12 in the final prime factorisation.

Step 1: Write (72=8\times9). Step 2: (8=2^3) and (9=3^2), so (72=2^3\times3^2). Step 3: 8 and 9 are composite, so they are not the final prime form.

Step 1: Changing the order in multiplication does not change the product. Step 2: For example, (2\times3\times5) and (5\times3\times2) both give 30. Step 3: In prime factorisation, the factors matter, not their order.

Step 1: Write (96=32\times3). Step 2: (32=2^5), so (96=2^5\times3). Step 3: Count repeated prime factors to write their powers.

Step 1: First evaluate the powers: (2^2=4) and (3^2=9). Step 2: (4\times9\times5=180), so the number is 180. Step 3: Simplify powers first, then multiply.

Step 1: Write (63=9\times7). Step 2: (9=3^2), so (63=3^2\times7). Step 3: 9 and 21 are composite, so do not keep them in the final form.

Step 1: A prime number has exactly two positive factors. Step 2: 31 is divisible only by 1 and 31, while 27, 35, and 45 are composite. Step 3: For small numbers, checking divisibility by 2, 3, 5, and 7 is useful.

Step 1: A composite number has more than two positive factors. Step 2: (77=7\times11), so it is composite. Step 3: If a number can be written as a product of two smaller primes, it is composite.

Step 1: Write (135=27\times5). Step 2: (27=3^3), so (135=3^3\times5). Step 3: 9 and 15 may form the product, but they are not prime factorisation.