Prime Factorisation

Step 1: (3^2=9). Step 2: (9 \times 5=45), so (m=45). Step 3: Keeping the order of powers and multiplication clear reduces mistakes.

Step 1: (216=8 \times 27). Step 2: (8=2^3) and (27=3^3), so (216=2^3 \times 3^3). Step 3: Recognising cube numbers is very useful in medium-level questions.

Step 1: (3^3=27). Step 2: (27 \times 7=189), so the number is (189). Step 3: Remembering small powers helps you calculate faster.

Step 1: Write (540=54 \times 10). Step 2: (54=2 \times 3^3) and (10=2 \times 5), so (540=2^2 \times 3^3 \times 5). Step 3: Splitting a large number into easy parts is a safe method.

Step 1: A divisor must not need prime exponents greater than those available. Step 2: (72=2^3 \times 3^2), which is fully present in (2^4 \times 3^2). Step 3: For divisibility, match the exponent of each prime separately.

Step 1: A trailing zero is formed by a pair (10=2 \times 5). Step 2: The exponent of (2) is (3) and of (5) is (2), so (2) pairs can be formed. Step 3: For trailing zeros, take the smaller exponent of (2) and (5).

Step 1: Write (120=12 \times 10). Step 2: (12=2^2 \times 3) and (10=2 \times 5), so (120=2^3 \times 3 \times 5). Step 3: Do not forget to combine repeated prime factors from different parts.

Step 1: An odd factor must not contain (2). Step 2: Removing (2^5) leaves only (3), so the greatest odd factor is (3). Step 3: For the greatest odd factor, remove all powers of (2).

Step 1: In a perfect cube, each prime exponent must be a multiple of (3). Step 2: To make (2^2) into (2^3) and (3^2) into (3^3), multiply by (2 \times 3=6). Step 3: For a cube, exponents should be like (3,6,9).

Step 1: In a perfect square, every prime exponent must be even. Step 2: To make (2^3) into (2^4) and (3) into (3^2), multiply by (2 \times 3=6). Step 3: For a square, increase odd exponents by one to make them even.

Step 1: Divide (64) repeatedly by (2). Step 2: (64=2 \times 2 \times 2 \times 2 \times 2 \times 2=2^6). Step 3: (4^3) gives the value, but it is not prime factorisation because (4) is not prime.

Step 1: Prime factorisation means writing a number as a product of prime numbers. Step 2: Therefore, the bases must be prime numbers only. Step 3: Treating (1) as a prime factor is a major mistake.

Step 1: Add (1) to each exponent for total factors. Step 2: ((2+1)(1+1)(1+1)=3 \times 2 \times 2=12). Step 3: Do not forget primes with exponent (1).

Step 1: Write (360=36 \times 10). Step 2: (36=2^2 \times 3^2) and (10=2 \times 5), so (360=2^3 \times 3^2 \times 5). Hence (a=3, b=2). Step 3: Add exponents of repeated prime factors.

Step 1: The bases in prime factorisation are the prime factors. Step 2: Here the bases are (2) and (3), and the smallest is (2). Step 3: To find the smallest prime factor, do not focus on exponents.

Step 1: Write (75=3 \times 25). Step 2: Since (25=5^2), (75=3 \times 5^2). Step 3: Do not leave (25) in the final answer because it is not prime.

Step 1: (10=2 \times 5). Step 2: The exponent of (2) is (4) and of (5) is (1), so only (1) pair of (10) can be formed. Step 3: The maximum number of divisions by (10) is decided by the smaller exponent.

Step 1: In a perfect square, all prime exponents are even. Step 2: Here every exponent is (2), so (n) is a perfect square. Step 3: To identify a perfect square, check whether exponents are even.

Step 1: In a perfect cube, all prime exponents are multiples of (3). Step 2: Both exponents are (3), so (n) is a perfect cube. Step 3: For a perfect cube, check exponents using (3).

Step 1: The prime factors are the base numbers. Step 2: Here (2,3,5) are prime factors, and the greatest is (5). Step 3: When the greatest prime factor is asked, do not choose a composite number.

Step 1: A factor divisible by (3) must have exponent of (3) at least (1). Step 2: The exponent of (2) has (3) choices (0,1,2), and exponent of (3) has (3) choices (1,2,3). Total (3 \times 3=9). Step 3: For conditional factors, adjust exponent choices carefully.

Step 1: A factor divisible by (10) must contain both (2) and (5). Step 2: The exponent of (2) can be (1) to (4), giving (4) choices, and the exponent of (5) can be (1) to (2), giving (2) choices. Total (4 \times 2=8). Step 3: Divisibility by (10) needs both prime factors.

Step 1: In a square factor, every prime exponent must be even. Step 2: For (2), choices are (0,2), so (2) choices; for (3), choices are (0,2), so (2) choices; for (5), only (0), so (1) choice. Total (2 \times 2 \times 1=4). Step 3: Count even exponent choices separately.

Step 1: Every integer greater than (1) has a unique prime factorisation apart from the order. Step 2: This comes from the fundamental theorem of arithmetic. Step 3: Remember uniqueness of prime factorisation with this theorem.

Step 1: A prime number has exactly two positive factors. Step 2: (1) has only one positive factor, so it is not prime and has no prime factor. Step 3: Do not make the mistake of treating (1) as prime.