Fundamental Theorem of Arithmetic

Step 1: When counting distinct prime factors, do not add exponents. Step 2: The prime bases are 2, 3, 5, 11, and 13. Step 3: Therefore, the number of distinct prime factors is 5.

Step 1: The prime factors of the first number are 2, 3, and 11. Step 2: The prime factors of the second number are 5, 7, and 13. Step 3: There is no common prime factor, so they are co-prime.

Step 1: Co-prime numbers have HCF 1. Step 2: For two numbers, product (=) HCF (\times) LCM. Step 3: Therefore, the LCM will be 4199.

Step 1: Divide 32768 repeatedly by 2. Step 2: (32768=2^{15}). Step 3: (4^8) and (16^4) may give values, but 4 and 16 are not prime.

Step 1: Recognise (176400=420^2). Step 2: Since (420=2^2\times3\times5\times7), (420^2=2^4\times3^2\times5^2\times7^2). Step 3: In a perfect square, every prime exponent is even.

Step 1: (20655=3^5\times5\times17). Step 2: All these prime factors are present in (n) with sufficient powers. Step 3: Therefore, (n) must be divisible by 20655.

Step 1: In (n), the powers are (2^8), (3^4), and (5^2). Step 2: (3375=3^3\times5^3), which needs power 3 of 5. Step 3: Since (n) has only (5^2), (n) is not divisible by 3375.

Step 1: Calculate (2^9=512) and (3^4=81). Step 2: (512\times81\times5=207360). Step 3: In larger products, simplifying powers first is the right method.

Step 1: Calculate (2^5=32), (3^6=729), and (5^3=125). Step 2: (32\times729\times125=2916000). Step 3: Simplifying powers first keeps the calculation manageable.

Step 1: In LCM, take the higher power of each prime. Step 2: The powers of 2 are 5 and 8. Step 3: The higher power is 8, so the answer is 8.

Step 1: In HCF, take the smaller power of the common prime. Step 2: The powers of 3 are 6 and 3. Step 3: The smaller power is 3, so the answer is 3.

Step 1: Write (131220=26244\times5). Step 2: (26244=4\times6561=2^2\times3^8). Step 3: Therefore, (131220=2^2\times3^8\times5).

Step 1: For a perfect cube, exponents must be multiples of 3. Step 2: Make (2^{10}), (3^5), and (5^4) into powers 12, 6, and 6. Step 3: The smallest multiplier is (2^2\times3\times5^2).

Step 1: (63504=16\times3969=2^4\times3^4\times7^2). Step 2: All exponents are even, so the number is already a perfect square. Step 3: The smallest divisor should be 1, but 1 is not listed; therefore no given option is correct.

Step 1: In multiplication, exponents of the same prime base are added. Step 2: The power of 7 in (x) is 4 and in (y) is 5. Step 3: In (xy), the power of 7 is (4+5=9).

Step 1: Powers of the same base 2 are added in multiplication. Step 2: The power of 2 in (x) is 8 and in (y) is 7. Step 3: The total power is (8+7=15).

Step 1: Write (510510=30030\times17). Step 2: (30030=2\times3\times5\times7\times11\times13). Step 3: Therefore, the distinct prime factors are 2, 3, 5, 7, 11, 13, and 17.

Step 1: The common prime factors are 2, 3, and 11. Step 2: The smaller powers are (2^4), (3^3), and (11^1). Step 3: (16\times27\times11=4752), so the HCF is 4752.

Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^6), (3^5), (5^2), and (11^2). Step 3: (64\times243\times25\times121=47044800), so the correct value is 47044800.

Step 1: The product is (180\times25200). Step 2: (180=2^2\times3^2\times5) and (25200=2^4\times3^2\times5^2\times7). Step 3: The power of 2 in the product is (2+4=6).