Prime Factorisation

Step 1: Prime factorise (108). Step 2: (108=4 \times 27=2^2 \times 3^3), so (a=2) and (b=3). Hence (a+b=5). Step 3: Identify both exponents separately first.

Step 1: (144=16 \times 9). Step 2: (16=2^4) and (9=3^2), so (144=2^4 \times 3^2). Step 3: Recognising square numbers helps in prime factorisation.

Step 1: Look at the exponent of (2) in each option. Step 2: The exponents are (2,4,3,1), and the greatest is (4). Step 3: For comparison, you need not calculate the whole number; check only the required exponent.

Step 1: Write (250=25 \times 10). Step 2: (25=5^2) and (10=2 \times 5), so (250=2 \times 5^3). Step 3: Combine repeated (5) factors into a power.

Step 1: For total factors, add (1) to each exponent and multiply. Step 2: ((2+1)(2+1)(1+1)=3 \times 3 \times 2=18). Step 3: If an exponent is not shown, take it as (1).

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.