Prime Factorisation

Step 1: Write (84) as (4 \times 21). Step 2: Since (4=2^2) and (21=3 \times 7), (84=2^2 \times 3 \times 7). Step 3: In exams, multiply back to verify the original number.

Step 1: Write (126) as (2 \times 63). Step 2: Since (63=3^2 \times 7), (126=2 \times 3^2 \times 7). Step 3: While counting distinct prime factors, do not count exponents as separate numbers.

Step 1: Evaluate the powers first. Step 2: (2^3=8) and (3^2=9), so (8 \times 9 \times 5=360). Step 3: In such questions, solve powers before multiplying.

Step 1: Split (150) as (15 \times 10). Step 2: (15=3 \times 5) and (10=2 \times 5), so (150=2 \times 3 \times 5^2). Step 3: Writing repeated prime factors as powers keeps the answer neat.

Step 1: Evaluate the given power. Step 2: (2^2 \times 3 \times 5=4 \times 3 \times 5=60). Step 3: Multiply completely before selecting the option.

Step 1: Write (96) as (32 \times 3). Step 2: Since (32=2^5), (96=2^5 \times 3). Step 3: To find an exponent, divide repeatedly by that prime.

Step 1: In prime factorisation, every base must be a prime number. Step 2: (9) is not prime because (9=3^2), so the second option is not prime factorisation. Step 3: Always check each base carefully.

Step 1: (2^4=16). Step 2: (16 \times 3=48), so the number is (48). Step 3: Evaluating the power first makes calculation easier.

Step 1: (225=15 \times 15). Step 2: Since (15=3 \times 5), (225=3^2 \times 5^2). Step 3: (15^2) shows the value, but it is not prime factorisation because (15) is not prime.

Step 1: Prime factorise (72). Step 2: (72=8 \times 9=2^3 \times 3^2), so (a=3). Step 3: For an unknown exponent, compare prime factorisations on both sides.

Step 1: Write (180=18 \times 10). Step 2: (18=2 \times 3^2) and (10=2 \times 5), so (180=2^2 \times 3^2 \times 5). Step 3: Focus on the required exponent instead of memorising the whole number.

Step 1: (2^3=8) and (5^2=25). Step 2: (8 \times 25=200), so the number is (200). Step 3: Solving powers first makes multiplication simple.

Step 1: Write (315=9 \times 35). Step 2: (9=3^2) and (35=5 \times 7), so (315=3^2 \times 5 \times 7). It does not include (2). Step 3: A prime not appearing in the factorisation is not a prime factor of the number.

Step 1: Distinct prime factors are identified by the prime bases. Step 2: The bases here are (2) and (3), so there are (2) distinct prime factors. Step 3: The sum of exponents and the count of distinct primes are different ideas.

Step 1: Write (400=4 \times 100). Step 2: (4=2^2) and (100=2^2 \times 5^2), so (400=2^4 \times 5^2). Step 3: (4) is not prime, so final factorisation must use only prime bases.

Step 1: An even number must have (2) in its prime factorisation. Step 2: Only the third option contains (2), so it forms an even number. Step 3: You do not need to calculate the whole number to check even or odd.

Step 1: An odd number does not have (2) as a prime factor. Step 2: The third option has (3,5,7) and no (2), so it forms an odd number. Step 3: If (2) appears, the number is even.

Step 1: Add the exponents to find the total frequency of prime factors. Step 2: The exponents are (2,1,2), so the total is (2+1+2=5). Step 3: When no exponent is written, treat it as (1).

Step 1: To count total factors, add (1) to each exponent. Step 2: ((3+1)(2+1)=4 \times 3=12). Step 3: Use this method when the number is in prime factorised form.

Step 1: Write (90=9 \times 10). Step 2: (9=3^2) and (10=2 \times 5), so (90=2 \times 3^2 \times 5). Step 3: Splitting into easy parts like (9) and (10) is quick.

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).