Prime Factorisation

Step 1: Write (210) as (21 \times 10). Step 2: (21=3 \times 7) and (10=2 \times 5), so (210=2 \times 3 \times 5 \times 7). Step 3: After factorisation, check that every base is prime.

Step 1: Write (168) as (8 \times 21). Step 2: (8=2^3) and (21=3 \times 7), so (168=2^3 \times 3 \times 7). Step 3: To find the required exponent, you do not need to calculate any extra value.

Step 1: Evaluate the powers first. Step 2: (2^2=4) and (3^3=27), so (4 \times 27 \times 5=540). Step 3: Solving powers before multiplication is a safe method.

Step 1: Write (196) as (14 \times 14). Step 2: Since (14=2 \times 7), (196=2^2 \times 7^2). Step 3: Do not leave composite bases like (4) or (14) in the final answer.

Step 1: Write (252) as (4 \times 63). Step 2: (4=2^2) and (63=3^2 \times 7), so (a=2) and (b=2). Hence (a+b=4). Step 3: Match prime bases to identify unknown exponents.

Step 1: To count total factors, add (1) to each exponent. Step 2: ((4+1)(2+1)=5 \times 3=15). Step 3: While counting factors, focus on the prime exponents.

Step 1: Total frequency is found by adding the exponents. Step 2: The exponents are (3,1,2), so the total is (3+1+2=6). Step 3: If no exponent is written, treat it as (1).

Step 1: Write (320) as (32 \times 10). Step 2: (32=2^5) and (10=2 \times 5), so (320=2^6 \times 5). Step 3: Do not keep (10) in the final form because it is not prime.

Step 1: Write (432) as (16 \times 27). Step 2: (16=2^4) and (27=3^3), so (432=2^4 \times 3^3). Step 3: Recognising squares and cubes makes factorisation faster.

Step 1: (2^2=4). Step 2: (4 \times 3 \times 11=12 \times 11=132). Step 3: Forming small products first reduces mistakes.

Step 1: Write (288) as (32 \times 9). Step 2: (32=2^5) and (9=3^2), so (288=2^5 \times 3^2). Step 3: Clearly identify the required exponent in the final answer.

Step 1: A trailing zero is made by one pair of (2) and (5). Step 2: The exponent of (2) is (5) and of (5) is (3), so (3) pairs are possible. Step 3: For trailing zeros, choose the smaller exponent.

Step 1: (2^4=16). Step 2: (16 \times 3 \times 5=16 \times 15=240). Step 3: Calculate the number yourself before checking the options.

Step 1: Write (175) as (25 \times 7). Step 2: Since (25=5^2), (175=5^2 \times 7). Step 3: (25 \times 7) gives the value, but it is not final prime factorisation.

Step 1: Distinct prime factors are counted from the prime bases. Step 2: The bases are (2,3,7), so there are (3) distinct prime factors. Step 3: Do not count exponents as separate prime factors.

Step 1: An odd number does not contain (2) in its prime factorisation. Step 2: The second option has (3) and (7) but no (2), so it is odd. Step 3: The presence of (2) makes a number even.

Step 1: An even number must have (2) as a prime factor. Step 2: Only the third option contains (2), so it represents an even number. Step 3: To check evenness, calculating the whole number is not necessary.

Step 1: Prime factors are the base numbers. Step 2: The prime bases here are (2,3,5), and the greatest is (5). Step 3: Do not treat a number like (9) as a prime factor.

Step 1: In a perfect cube, every prime exponent must be a multiple of (3). Step 2: (2^4) needs (2^2) to become (2^6), and (3^2) needs (3) to become (3^3). The least multiplier is (12). Step 3: Since (12) is not in the options, the listed choices contain an error.

Step 1: In a perfect square, all prime exponents must be even. Step 2: (2^2) is already even, and (3^3) needs one more (3) to become (3^4). Step 3: Multiply only by the prime that has an odd exponent.

Step 1: A factor divisible by (3) must have exponent of (3) at least (1). Step 2: The exponent of (2) has (6) choices from (0) to (5), and the exponent of (3) has (2) choices (1,2). Total (6 \times 2=12). Step 3: In condition-based questions, adjust exponent limits carefully.

Step 1: A factor divisible by (10) must contain at least one (2) and one (5). Step 2: Exponent choices for (2) are (1,2,3), and for (5) are (1,2,3). Total (3 \times 3=9). Step 3: Divisibility by (10) needs both primes.

Step 1: In a square factor, every prime exponent must be even. Step 2: For (2), choices are (0,2,4), so (3); for (3), choices are (0,2), so (2); for (5), only (0), so (1). Total (3 \times 2 \times 1=6). Step 3: Count even exponent choices separately.

Step 1: (625=25 \times 25). Step 2: Since (25=5^2), (625=5^2 \times 5^2=5^4). Step 3: You can also divide repeatedly by (5) to find the exponent.

Step 1: In prime factorisation, every base must be prime. Step 2: (15) is not prime because (15=3 \times 5), so the third option is not prime factorisation. Step 3: Identify hidden composite numbers in the options.