Fundamental Theorem of Arithmetic

Step 1: For a perfect cube, every exponent must be a multiple of 3. Step 2: We must make 10 to 12, 8 to 9, 5 to 6, and 1 to 3. Step 3: So the smallest multiplier is (2^2\times3\times5\times7^2).

Step 1: In multiplication, exponents of the same prime base are added. Step 2: The power of 3 in (a) is 6 and in (b) is 8. Step 3: In (ab), the power of 3 is (6+8=14).

Step 1: In (xy), powers with the same base 5 are added. Step 2: The power of 5 in (x) is 7 and in (y) is 5. Step 3: The total power is (7+5=12).

Step 1: HCF uses the smaller powers of common prime factors only. Step 2: The common factors are 2 and 3, with smaller powers (2^9) and (3^8). Step 3: So the correct form is (2^9\times3^8).

Step 1: LCM uses the highest powers of all prime factors. Step 2: The highest powers are (2^{12}), (3^{10}), (5^4), and (7^5). Step 3: So the correct form is (2^{12}\times3^{10}\times5^4\times7^5).

Step 1: Product of the two numbers is (120\times9240=1108800). Step 2: One number is 840, so the other is (1108800\div840=1320). Step 3: As a check, the HCF of 840 and 1320 is 120.

Step 1: Product of the two numbers is (144\times10080=1451520). Step 2: The other number is (1451520\div720=2016). Step 3: To check, the HCF of 720 and 2016 is 144.

Step 1: Write (136080=3888\times35). Step 2: (3888=2^4\times3^5) and (35=5\times7), so (p=2^4\times3^5\times5\times7). Step 3: Comparing gives (a=4).

Step 1: Write (1814400=18144\times100). Step 2: (18144=2^5\times3^4\times7) and (100=2^2\times5^2), so the actual form is (2^7\times3^4\times5^2\times7). Step 3: The power of 3 is (b=4).

Step 1: To count with repetition, add the exponents. Step 2: (2^9) gives 9, (3^7) gives 7, (5^6) gives 6, and (17^2) gives 2 factors. Step 3: Total (9+7+6+2=24), so the answer is 24.

Step 1: When counting distinct prime factors, exponents are not added. Step 2: The prime bases are 2, 3, 5, 7, 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 13. Step 2: The prime factors of the second number are 5, 7, and 11. 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 7429.

Step 1: Divide 65536 repeatedly by 2. Step 2: (65536=2^{16}). Step 3: (4^8) and (16^4) can give the value, but 4 and 16 are not prime.

Step 1: Recognise (705600=840^2). Step 2: Since (840=2^3\times3\times5\times7), (840^2=2^6\times3^2\times5^2\times7^2). Step 3: In a perfect square, all prime exponents are even.

Step 1: (69255=3^6\times5\times19). Step 2: All these prime factors are present in (n) with sufficient powers. Step 3: Therefore, (n) must be divisible by 69255.

Step 1: In (n), the powers are (2^9), (3^5), and (5^3). Step 2: (16875=3^3\times5^4), which needs power 4 of 5. Step 3: Since (n) has only (5^3), (n) is not divisible by 16875.

Step 1: Calculate (2^{10}=1024) and (3^4=81). Step 2: (1024\times81\times5=414720). Step 3: In larger multiplication, simplify powers first.

Step 1: Calculate (2^6=64), (3^6=729), and (5^3=125). Step 2: (64\times729\times125=5832000). Step 3: Simplifying powers first keeps the calculation clean.

Step 1: In LCM, take the higher power of each prime. Step 2: The powers of 2 are 6 and 9. Step 3: The higher power is 9, so the answer is 9.