Fundamental Theorem of Arithmetic

Correct answer: A. They will differ only in order

Explanation: Step 1: The Fundamental Theorem of Arithmetic states uniqueness of prime factorisation. Step 2: The order of prime factors may change, but the prime factors themselves do not change. Step 3: In exams, do not treat a change of order as a different factorisation.

Correct answer: A. (2^3\times3^2\times5\times7\times11)

Explanation: Step 1: Write (27720=2520\times11). Step 2: Since (2520=2^3\times3^2\times5\times7), the full factorisation is (2^3\times3^2\times5\times7\times11). Step 3: Do not leave a composite factor like 2520 in the final answer.

Correct answer: A. 108

Explanation: Step 1: For HCF, take the smaller powers of only the common prime factors. Step 2: The common prime factors are 2 and 3, with smaller powers (2^2) and (3^3). Step 3: (2^2\times3^3=4\times27=108), so the answer is 108.

Correct answer: B. 680400

Explanation: Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^4), (3^5), (5^2), and (7). Step 3: (16\times243\times25\times7=680400), so the answer is 680400.

Correct answer: B. 840

Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: Therefore, LCM (=60480\div72=840). Step 3: Use this relation directly only for two numbers.

Correct answer: A. 56700

Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (45\times1260=56700). Step 3: In such questions, first notice that exactly two numbers are involved.

Correct answer: A. 2

Explanation: Step 1: (1800=18\times100=2^3\times3^2\times5^2). Step 2: For a perfect square, all exponents must be even, but the exponent of 2 is 3. Step 3: Multiplying by 2 makes it 4, so the smallest number is 2.

Correct answer: C. 35

Explanation: Step 1: (8820=2^2\times3^2\times5\times7^2). Step 2: For a perfect square, all exponents must be even, but the exponent of 5 is 1. Step 3: Dividing by 5 makes all remaining exponents even, so the smallest number is 5.

Correct answer: A. 10

Explanation: Step 1: (5400=54\times100=2^3\times3^3\times5^2). Step 2: For a perfect cube, each prime exponent must be a multiple of 3. Step 3: The exponent of 5 is 2, so one more 5 is needed; the smallest number is 5.

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

Explanation: Step 1: Recognise (9261=21^3). Step 2: Since (21=3\times7), (21^3=3^3\times7^3). Step 3: (21^3) gives the value, but 21 is not prime, so write the final prime form separately.

Correct answer: A. 14

Explanation: Step 1: For a perfect square, every exponent must be even. Step 2: The exponents of (2^5) and (7^1) are odd, while the others are even. Step 3: Multiplying by (2\times7=14) makes all exponents even.

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

Explanation: Step 1: For a perfect cube, every exponent must be a multiple of 3. Step 2: To make powers (4,5,2) into (6,6,3), we need (2^2), (3), and (5). Step 3: So the smallest multiplier is (2^2\times3\times5).

Correct answer: C. 6

Explanation: Step 1: In multiplication, exponents of the same prime base are added. Step 2: The power of 3 in (a) is 2 and in (b) is 4. Step 3: In (ab), the power of 3 will be (2+4=6).

Correct answer: B. 5

Explanation: Step 1: In (xy), the powers of the same base 5 are added. Step 2: The power of 5 in (x) is 1 and in (y) is 4. Step 3: The total power is (1+4=5).

Correct answer: A. (2^5\times3^4)

Explanation: Step 1: HCF uses only common prime factors. Step 2: The common factors are 2 and 3; the smaller powers are (2^5) and (3^4). Step 3: Therefore, the HCF is (2^5\times3^4).

Correct answer: A. (2^8\times3^6\times5^2\times7)

Explanation: Step 1: LCM uses the highest powers of all prime factors. Step 2: The highest powers are (2^8), (3^6), (5^2), and (7). Step 3: So the correct form is (2^8\times3^6\times5^2\times7).

Correct answer: A. 108

Explanation: Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: Product of the two numbers is (18\times540=9720). Step 3: The other number is (9720\div90=108).

Correct answer: A. 168

Explanation: Step 1: Product of the two numbers is (24\times840=20160). Step 2: One number is 120, so the other is (20160\div120=168). Step 3: You can check the answer by confirming that HCF of 120 and 168 is 24.

Correct answer: C. 4

Explanation: Step 1: Prime factorise 720. Step 2: (720=72\times10=2^4\times3^2\times5). Step 3: Comparing with the given form gives (a=4).

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: Comparing gives (b=3).

Correct answer: C. 9

Explanation: Step 1: To count with repetition, add the exponents. Step 2: (2^4) gives 4, (3^3) gives 3, and (5^2) gives 2 factors. Step 3: Total (4+3+2=9), so the answer is 9.

Correct answer: A. 3

Explanation: Step 1: When counting distinct prime factors, do not add exponents. Step 2: The prime factors here are 2, 3, and 7. Step 3: Therefore, the number of distinct prime factors is 3.

Correct answer: A. They are co-prime

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

Correct answer: D. 1001

Explanation: Step 1: Co-prime numbers have HCF 1. Step 2: Product (=) HCF (\times) LCM. Step 3: Therefore, the LCM will be (1001).

Correct answer: A. (2^{12})

Explanation: Step 1: Divide 4096 repeatedly by 2. Step 2: (4096=2^{12}). Step 3: (4^6) and (16^3) can give the value, but 4 and 16 are not prime.