Prime Factorisation

Step 1: Write (132) as (12 \times 11). Step 2: (12=2^2 \times 3), so (132=2^2 \times 3 \times 11). Step 3: Always check that all bases in the final answer are prime.

Step 1: Write (224) as (32 \times 7). Step 2: (32=2^5), so (224=2^5 \times 7). Step 3: Repeated division by the same prime helps find its exponent.

Step 1: First evaluate (2^3). Step 2: (2^3=8), so (8 \times 3 \times 7=168). Step 3: Multiplying after evaluating powers keeps the calculation clear.

Step 1: Write (275) as (25 \times 11). Step 2: Since (25=5^2), (275=5^2 \times 11). Step 3: Do not leave (25) in the final form because it is not prime.

Step 1: Write (450) as (45 \times 10). Step 2: (45=3^2 \times 5) and (10=2 \times 5), so (450=2 \times 3^2 \times 5^2). Thus (a=1, b=2), so (a+b=3). Step 3: Combine like prime factors when finding unknown exponents.

Step 1: To count total factors, add (1) to each exponent and multiply. Step 2: ((3+1)(2+1)(1+1)=4 \times 3 \times 2=24). Step 3: Do not forget that an unwritten exponent is (1).

Step 1: Total frequency is found by adding the exponents. Step 2: The exponents are (4,1,2), so the total is (4+1+2=7). Step 3: If no exponent is shown, take it as (1).

Step 1: Write (384) as (128 \times 3). Step 2: (128=2^7), so (384=2^7 \times 3). Step 3: Remembering powers of (2) saves time in such questions.

Step 1: Write (675) as (27 \times 25). Step 2: (27=3^3) and (25=5^2), so (675=3^3 \times 5^2). Step 3: Splitting a number into familiar squares and cubes is a good method.

Step 1: (2^3=8). Step 2: (8 \times 13=104), so (n=104). Step 3: First evaluate the small power, then multiply.

Step 1: Write (324) as (4 \times 81). Step 2: (4=2^2) and (81=3^4), so (324=2^2 \times 3^4). Step 3: Identify the required exponent separately in the final answer.

Step 1: A trailing zero is formed by a pair of (2) and (5). Step 2: The exponent of (2) is (6) and of (5) is (4), so (4) pairs are possible. Step 3: For trailing zeros, always take the smaller exponent.

Step 1: (2^2=4) and (3^2=9). Step 2: (4 \times 9 \times 11=36 \times 11=396). Step 3: Evaluating powers first makes the calculation simple.

Step 1: Write (245) as (5 \times 49). Step 2: Since (49=7^2), (245=5 \times 7^2). Step 3: Do not leave (49) in the final answer because it is not prime.

Step 1: Distinct prime factors are counted from the prime bases. Step 2: The bases are (2,3,5), so there are (3) distinct prime factors. Step 3: Exponents are not counted as separate prime factors.

Step 1: An odd number does not contain (2) as a prime factor. Step 2: The second option has (3) and (5), but no (2), so it is odd. Step 3: The presence of (2) makes the number even.

Step 1: An even number must contain (2) in its prime factorisation. Step 2: Only the fourth option contains (2), so it represents an even number. Step 3: You do not need to calculate the whole number to check evenness.

Step 1: Prime factors are the base numbers. Step 2: The prime bases here are (2,3,11), and the greatest is (11). Step 3: Do not treat a composite value like (9) as a prime factor.

Step 1: In a perfect cube, every prime exponent must be a multiple of (3). Step 2: (2^5) needs one (2) to become (2^6), and (3^2) needs one (3) to become (3^3). So the multiplier is (2 \times 3=6). Step 3: Raise each exponent to the next multiple of (3).

Step 1: In a perfect square, every prime exponent must be even. Step 2: (2^4) is fine, while (3^3) needs one (3) and (5) needs one (5), so the multiplier is (15). Step 3: Multiply only by primes with odd exponents.

Step 1: A factor divisible by (3) must have the exponent of (3) at least (1). Step 2: The exponent of (2) has (5) choices from (0) to (4), and the exponent of (3) has (3) choices (1,2,3). Total (5 \times 3=15). Step 3: In conditional factor counting, adjust exponent limits.

Step 1: A factor divisible by (10) must contain both (2) and (5). Step 2: The exponent of (2) has (4) choices from (1) to (4), and the exponent of (5) has (3) choices from (1) to (3). Total (4 \times 3=12). 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), giving (3) choices; for (3), choices are (0,2), giving (2); for (7), only (0), giving (1). Total (6). Step 3: Count even exponent choices separately.

Step 1: (729=27 \times 27). Step 2: Since (27=3^3), (729=3^3 \times 3^3=3^6). Step 3: For larger powers, split the number into familiar cubes.

Step 1: In prime factorisation, every base must be prime. Step 2: (21) is not prime because (21=3 \times 7), so the third option is not prime factorisation. Step 3: Always identify hidden composite numbers in options.