Fundamental Theorem of Arithmetic

Correct answer: A. The order of factors may change but the prime factors remain the same

Explanation: Step 1: The theorem says that prime factorisation of a number greater than 1 is fixed. Step 2: Changing the order does not change the product, such as (2\times3\times5) and (5\times2\times3). Step 3: In exams, do not treat changed order as a different factorisation.

Correct answer: A. (2^2\times3^3\times5)

Explanation: Step 1: Write (540=54\times10). Step 2: (54=2\times3^3) and (10=2\times5), so (540=2^2\times3^3\times5). Step 3: Do not leave composite factors like 54 and 10 in the final form.

Correct answer: A. (2^2\times3^3\times7)

Explanation: Step 1: Write (756=84\times9). Step 2: (84=2^2\times3\times7) and (9=3^2), so (756=2^2\times3^3\times7). Step 3: Combine repeated prime factors using powers.

Correct answer: A. (2^2\times5\times7^2)

Explanation: Step 1: Write (980=98\times10). Step 2: (98=2\times7^2) and (10=2\times5), so (980=2^2\times5\times7^2). Step 3: Composite factors like 98 and 10 should not remain in the final prime form.

Correct answer: A. 36

Explanation: Step 1: For HCF, take the smaller powers of common prime factors. Step 2: The common factors are 2 and 3, with smaller powers (2^2) and (3^2). Step 3: (2^2\times3^2=4\times9=36), so the answer is 36.

Correct answer: B. 15120

Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^4), (3^3), (5), and (7). Step 3: (16\times27\times5\times7=15120), so the answer is 15120.

Correct answer: B. 84

Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (1512=18\times) LCM, so LCM (=1512\div18=84). Step 3: Use this relation directly only for two numbers.

Correct answer: B. 3150

Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (15\times210=3150). Step 3: When the question has two numbers, this formula gives the answer quickly.

Correct answer: B. 504

Explanation: Step 1: First calculate (2^3=8) and (3^2=9). Step 2: (8\times9\times7=504). Step 3: To get the number from prime factorisation, simplify powers first.

Correct answer: C. 3

Explanation: Step 1: Write (1512=8\times189). Step 2: (8=2^3) and (189=3^3\times7), so (1512=2^3\times3^3\times7). Step 3: The number of times 3 appears as a factor is its power.

Correct answer: A. (3^3\times5\times7)

Explanation: Step 1: Write (945=9\times105). Step 2: (9=3^2) and (105=3\times5\times7), so (945=3^3\times5\times7). Step 3: Do not keep composite factors like 9 or 105 in the final answer.

Correct answer: C. 360

Explanation: Step 1: Calculate (2^3=8) and (3^2=9). Step 2: (8\times9\times5=360). Step 3: When converting prime factorisation into a number, multiply all factors.

Correct answer: B. 3

Explanation: Step 1: Look at distinct prime factors, not powers. Step 2: In (m), the distinct prime factors are 2, 3, and 5. Step 3: In (2^4), count 2 only once in the distinct list.

Correct answer: A. They are co-prime

Explanation: Step 1: (16=2^4) and (45=3^2\times5). Step 2: They have no common prime factor, so they are co-prime. Step 3: Co-prime numbers have HCF 1.

Correct answer: D. 221

Explanation: Step 1: Co-prime numbers have HCF 1. Step 2: For two numbers, product (=) HCF (\times) LCM. Step 3: Therefore, the LCM is (221).

Correct answer: A. (2^2\times3\times11)

Explanation: Step 1: In prime factorisation, every factor must be prime. Step 2: (2), (3), and (11) are prime, so (2^2\times3\times11) is correct. Step 3: 4, 6, and 12 are composite, so they should not remain in the final form.

Correct answer: B. 24

Explanation: Step 1: The common prime factors are 2 and 3. Step 2: The smaller powers are (2^3) and (3^1), so (8\times3=24). Step 3: For HCF, take smaller powers of only common factors.

Correct answer: C. 360

Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^3), (3^2), and (5^1). Step 3: (8\times9\times5=360), so the answer is 360.

Correct answer: A. (2^2\times3^2\times5\times7)

Explanation: Step 1: Write (1260=126\times10). Step 2: (126=2\times3^2\times7) and (10=2\times5). Step 3: Therefore, (1260=2^2\times3^2\times5\times7).

Correct answer: B. 288

Explanation: Step 1: Calculate (2^5=32) and (3^2=9). Step 2: (32\times9=288). Step 3: In prime factorisation with powers, evaluate powers first.

Correct answer: C. 7

Explanation: Step 1: When repetition is counted, add the exponents. Step 2: (2^4) gives 4 factors, (3^2) gives 2 factors, and (5) gives 1 factor. Step 3: Total (4+2+1=7), so the answer is 7.

Correct answer: B. 2

Explanation: Step 1: Prime factorise 108. Step 2: (108=4\times27=2^2\times3^3), so the given form (2^a\times3^2) does not match 108. Step 3: The power of 3 should also be 3; therefore no value of (a) alone can make it correct.

Correct answer: C. 3

Explanation: Step 1: Write 216 as (8\times27). Step 2: (8=2^3) and (27=3^3), so (216=2^3\times3^3). Step 3: Therefore, (a=3).

Correct answer: A. They are co-prime

Explanation: Step 1: The prime factors of the first number are 2 and 5. Step 2: The prime factors of the second number are 3 and 7. Step 3: There is no common prime factor, so they are co-prime.

Correct answer: C. 630

Explanation: Step 1: (18=2\times3^2) and (35=5\times7), so they are co-prime. Step 2: The LCM of co-prime numbers equals their product. Step 3: (18\times35=630), so the answer is 630.