Prime Factorisation

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.

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, every prime exponent is a multiple of (3). Step 2: Both (6) and (3) are multiples of (3), so (N) is a perfect cube. Step 3: For a perfect cube, check exponents using (3).

Step 1: An odd factor must not contain (2). Step 2: After removing (2^2), we get (3^3 \times 5=27 \times 5=135). Step 3: To get the greatest odd factor, remove all powers of (2).

Step 1: The given number has no (2), so it is odd. Step 2: To make the smallest even multiple, multiplying by one (2) is enough. Step 3: When the smallest multiple is asked, do not increase exponents unnecessarily.

Step 1: The square root of a number is an integer only when all prime exponents are even. Step 2: In (2^5 \times 3), the exponents are (5) and (1), both odd. Step 3: For square-root questions, check evenness of exponents first.