Prime Factorisation

Step 1: Write (518400=256\times2025). Step 2: (256=2^8) and (2025=3^4\times5^2), so (518400=2^8\times3^4\times5^2). Step 3: Give 2025 its final prime form.

Step 1: (529200=16\times33075). Step 2: (33075=3^3\times5^2\times7^2), so the power of 7 is 2. Step 3: Comparing gives (q=2).

Step 1: Write (627264=64\times9801). Step 2: (64=2^6) and (9801=3^4\times11^2), so the power of 2 is 6. Step 3: From the given form, (a=6).

Step 1: (680400=16\times42525). Step 2: (42525=3^5\times5^2\times7), so the power of 3 is 5. Step 3: Comparing gives (b=5).

Step 1: Write (137200=16\times8575). Step 2: (8575=5^2\times7^3), so the power of 7 is 3. Step 3: From the given form, (p=3).

Step 1: Calculate (2^6=64), (3^5=243), and (7^2=49). Step 2: (64\times243\times49=762048). Step 3: In such calculations, solve powers first.

Step 1: Calculate (2^4=16), (3^5=243), and (7^2=49). Step 2: (16\times243\times49=190512). Step 3: Finding all three powers separately is safer.

Step 1: Calculate (2^7=128), (3^2=9), and (11^2=121). Step 2: (128\times9\times121=139392). Step 3: Find the values of powers first, then multiply.

Step 1: Calculate (3^4=81), (5^2=25), and (7^2=49). Step 2: (81\times25\times49=99225). Step 3: Do not skip square powers in a hurry.

Step 1: Calculate (2^3=8) and (3^6=729). Step 2: (8\times729\times5\times13=379080). Step 3: Simplify the higher-power part first.

Step 1: In a perfect square, all exponents are even. Step 2: The powers of 2 and 11 are odd. Step 3: Multiplying by (2\times11=22) makes both powers even.

Step 1: For a perfect square, exponents should be even. Step 2: The powers of 3 and 7 are odd. Step 3: Dividing by (3\times7=21) makes the powers 4 and 2.

Step 1: In a perfect cube, every exponent must be a multiple of 3. Step 2: Powers 5, 4, 2, and 7 must become 6, 6, 3, and 9. Step 3: The smallest multiplier is (2\times3^2\times5\times7^2).

Step 1: In a perfect cube, exponents are multiples of 3. Step 2: Reducing 10 to 9, 8 to 6, 5 to 3, and 4 to 3 is the smallest way. Step 3: Therefore, the divisor is (2\times3^2\times5^2\times11).

Step 1: Every prime power of a divisor must be available in the number. Step 2: (n) has power 2 of 5, but (5^3) needs power 3. Step 3: Therefore, (n) is not divisible by (5^3).

Step 1: For divisibility, the exponents of the divisor must not exceed those in the number. Step 2: In the first option, all exponents are less than or equal to those in (n). Step 3: Therefore, (n) must be divisible by that number.

Step 1: To count with repetition, add the exponents. Step 2: (8+6+4+3+2=23). Step 3: Counting only bases and counting with repetition are different.

Step 1: While counting distinct primes, only bases are counted. Step 2: The bases are 2, 3, 5, 7, 11, and 13. Step 3: Therefore, the number of distinct prime factors is 6.

Step 1: In multiplication, powers of the same prime base are added. Step 2: The power of 11 in (a) is 1 and in (b) is 2. Step 3: In (ab), the power of 11 will be (1+2=3).

Step 1: Powers with the same base 3 are added in multiplication. Step 2: The power of 3 in (x) is 4 and in (y) is 5. Step 3: The total power will be (4+5=9).