Fundamental Theorem of Arithmetic

Step 1: This theorem is about prime factorisation. Step 2: Every composite number can be written as a product of prime numbers. Step 3: In exams, remember this theorem through factorisation.

Step 1: Break 36 into small prime factors. Step 2: (36=2\times2\times3\times3=2^2\times3^2). Step 3: In the final answer, keep only prime numbers as factors.

Step 1: Write 60 as (2\times30), then (30=2\times3\times5). Step 2: So (60=2^2\times3\times5). Step 3: (4\times15) gives the product, but it is not prime factorisation.

Step 1: Uniqueness means the set of prime factors is fixed. Step 2: The order may change, such as (2\times3) and (3\times2), but the factors remain the same. Step 3: Do not get confused by the order of factors.

Step 1: (84=2\times42=2\times2\times21). Step 2: So (84=2^2\times3\times7), where the power of 2 is 2. Step 3: To find the power, count repeated prime factors.

Step 1: First find the powers: (2^3=8) and (3^2=9). Step 2: (8\times9=72), so the number is 72. Step 3: Calculate powers first, then multiply.

Step 1: (45=9\times5) and (9=3\times3). Step 2: Therefore, (45=3^2\times5). Step 3: Composite factors like 9 or 15 are not kept in final prime factorisation.

Step 1: A prime number has exactly two positive factors, 1 and itself. Step 2: 29 is divisible only by 1 and 29, while the other numbers 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: (49=7\times7), so it has a factor 7 besides 1 and 49. Step 3: Recognising perfect squares helps in finding composite numbers.

Step 1: Write (90=9\times10). Step 2: (9=3^2) and (10=2\times5), so (90=2\times3^2\times5). Step 3: In the final form, all factors must be prime.

Step 1: Look at distinct prime factors, not their powers. Step 2: In (72=2^3\times3^2), the distinct primes are 2 and 3. Step 3: The number of distinct prime factors and total prime factors are different.

Step 1: Write (100=10\times10). Step 2: Each 10 is (2\times5), so (100=2^2\times5^2). Step 3: 4, 10, and 25 are composite, so do not keep them in the final prime form.

Step 1: For HCF, take the smaller powers of common prime factors. Step 2: The smaller power of (2) is 1 and of (3) is 1, so (2\times3=6). Step 3: Use smaller powers for HCF.

Step 1: For LCM, take the highest powers of all prime factors. Step 2: The higher power of (2) is 2 and of (3) is 2, so (2^2\times3^2=36). Step 3: Use higher powers for LCM.

Step 1: Write (144=16\times9). Step 2: (16=2^4) and (9=3^2), so (144=2^4\times3^2). Step 3: 12 is composite, so (12\times12) is not prime factorisation.

Step 1: First find (3^2=9). Step 2: (9\times5=45), so the number is 45. Step 3: In factorisation with powers, evaluate the power first.

Step 1: Prime factorisation means writing a number as a product of prime numbers. Step 2: So the final factors should be primes such as 2, 3, 5, and 7. Step 3: If a composite factor remains, factorise it further.

Step 1: Write (75=3\times25). Step 2: (25=5\times5=5^2), so (75=3\times5^2). Step 3: 15 and 25 are composite, so do not keep them in the final form.

Step 1: Write (56=8\times7). Step 2: (8=2^3), so (56=2^3\times7). Step 3: The distinct prime factors are 2 and 7.

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: Co-prime numbers have no common prime factor. Step 2: (8=2^3) and (15=3\times5), so they have no common prime factor. Step 3: If a common factor appears, the pair is not co-prime.

Step 1: The common prime factors are 2 and 5. Step 2: The smaller powers are (2^1) and (5^1), so the HCF is (2\times5=10). Step 3: Remember to take smaller powers for HCF.

Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^3) and (5^2), so (8\times25=200). Step 3: LCM uses the highest powers.

Step 1: Write (120=12\times10). Step 2: (12=2^2\times3) and (10=2\times5), so (120=2^3\times3\times5). Step 3: Write repeated prime factors using powers.

Step 1: Find (2^2=4). Step 2: (4\times7=28), so the number is 28. Step 3: To get the number from prime factorisation, simplify powers first.