Fundamental Theorem of Arithmetic

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.

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.