Prime Factorisation

Correct answer: A. 24

Explanation: Step 1: In prime factorisation, add 1 to each exponent and multiply. Step 2: Here the number of factors is ((3+1)(2+1)(1+1)=24). Step 3: In exams, identify the exponents first and then multiply carefully.

Correct answer: A. (2)

Explanation: Step 1: The bases in prime factorisation are the prime factors. Step 2: Among (2,3,7), the smallest prime number is (2). Step 3: For the smallest prime factor, look at the base, not the exponent.

Correct answer: A. 300

Explanation: Step 1: Evaluate the powers first. Step 2: (2^2 \times 3 \times 5^2=4 \times 3 \times 25=300). Step 3: In such questions, solve powers before multiplying.

Correct answer: A. (2)

Explanation: Step 1: In HCF, take the smaller exponent of common prime factors. Step 2: The exponents of (3) are (2) and (4), so the smaller one is (2). Step 3: For HCF, always choose the minimum exponent.

Correct answer: A. (4)

Explanation: Step 1: In LCM, take the larger exponent of each prime factor. Step 2: The exponents of (5) are (2) and (4), so the larger one is (4). Step 3: Remember that LCM uses maximum exponents.

Correct answer: A. (2^3 \times 3^2 \times 5)

Explanation: Step 1: Write (360=36 \times 10). Step 2: Since (36=2^2 \times 3^2) and (10=2 \times 5), (360=2^3 \times 3^2 \times 5). Step 3: Verify by multiplying back to the original number.

Correct answer: A. (3)

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

Correct answer: A. 3

Explanation: Step 1: Distinct prime factors are counted by their prime bases. Step 2: The bases are (2,3,5), so there are (3) distinct prime factors. Step 3: Do not count exponents as separate factors.

Correct answer: A. 336

Explanation: Step 1: (2^4 \times 3 \times 7=16 \times 21=336). Step 2: (336,672,1008,1344) are all divisible by (336), so none is actually not divisible. Step 3: In exams, read negative wording and test every option carefully.

Correct answer: A. 264

Explanation: Step 1: Evaluate the power first. Step 2: (2^3=8), so (N=8 \times 3 \times 11=264). Step 3: Group smaller products to calculate faster.

Correct answer: A. 400

Explanation: Step 1: The condition requires (2^4 \times 5^2). Step 2: (2^4 \times 5^2=16 \times 25=400), and it has no factor (3). Step 3: Check the exponent of every prime before choosing.

Correct answer: A. (1)

Explanation: Step 1: Write (840=84 \times 10). Step 2: Since (84=2^2 \times 3 \times 7) and (10=2 \times 5), (840=2^3 \times 3 \times 5 \times 7). The exponent of (7) is (1). Step 3: A prime appearing once has exponent (1).

Correct answer: A. (45)

Explanation: Step 1: A divisor must not require prime exponents higher than those available. Step 2: (45=3^2 \times 5), which is fully contained in (3^2 \times 5^3). Step 3: Compare exponents to test divisibility.

Correct answer: A. 12

Explanation: Step 1: For a square factor, every prime exponent must be even. Step 2: For (2^3), even choices are (0,2); for (3^2), (0,2); for (5^2), (0,2). Total (2 \times 2 \times 2=8). Step 3: Count only even exponent choices for square factors.

Correct answer: A. (2^2 \times 3^3 \times 5)

Explanation: 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 number into two easy parts saves time.

Correct answer: A. 156

Explanation: Step 1: First calculate (2^2=4). Step 2: (4 \times 3 \times 13=12 \times 13=156). Step 3: Break multiplication into smaller steps.

Correct answer: A. (3)

Explanation: Step 1: In a perfect cube, every prime exponent must be a multiple of (3). Step 2: (2^6) is fine, but (3^2) needs one more (3) to become (3^3). Step 3: For cubes, make exponents multiples of (3).

Correct answer: A. (2)

Explanation: Step 1: In a perfect square, all prime exponents are even. Step 2: (2^5) has an odd exponent, so multiplying by (2) makes it (2^6); (7^2) is already even. Step 3: For squares, fix only the odd exponents.

Correct answer: A. 2

Explanation: Step 1: One pair of (2) and (5) makes (10), giving a trailing zero. Step 2: Remove one (2) with the (5), leaving (2^3 \times 3^2=72), so the last non-zero digit is (2). Step 3: After removing trailing-zero pairs, check the last digit of the remaining product.

Correct answer: A. 1

Explanation: Step 1: A trailing zero comes from a pair (10=2 \times 5). Step 2: The exponent of (2) is (3) and of (5) is (1), so there is (1) pair. Step 3: For trailing zeros, take the smaller exponent of (2) and (5).

Correct answer: A. (11)

Explanation: Step 1: Write (1260=126 \times 10). Step 2: (126=2 \times 3^2 \times 7) and (10=2 \times 5), so (1260=2^2 \times 3^2 \times 5 \times 7). It does not include (11). Step 3: Match the options with the prime factorisation.

Correct answer: A. (2^2 \times 3^2 \times 5^2)

Explanation: Step 1: Write (900=9 \times 100). Step 2: (9=3^2) and (100=2^2 \times 5^2), so (900=2^2 \times 3^2 \times 5^2). Step 3: Using square numbers makes factorisation easier.

Correct answer: A. Perfect square

Explanation: Step 1: In a perfect square, every prime exponent is even. Step 2: Here all exponents are (2), so (N) is a perfect square. Step 3: To identify a square, check whether all exponents are even.

Correct answer: A. Perfect cube

Explanation: Step 1: In a perfect cube, all prime exponents are multiples of (3). Step 2: Each exponent is (3), so (N) is a perfect cube. Step 3: To identify a cube, check divisibility of exponents by (3).

Correct answer: A. 6

Explanation: Step 1: (144=12^2). Step 2: Since (12=2^2 \times 3), (144=2^4 \times 3^2); the sum is (4+2=6). Step 3: When squaring a number, its prime exponents double.