Fundamental Theorem of Arithmetic

Step 1: When counting distinct prime factors, exponents are not added. Step 2: The prime bases are 2, 3, 5, 7, and 17. 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 12673.

Step 1: Divide 131072 repeatedly by 2. Step 2: (131072=2^{17}). Step 3: (4^9) and (16^5) are not prime factorisations because 4 and 16 are composite.

Step 1: Recognise (2822400=1680^2). Step 2: Since (1680=2^4\times3\times5\times7), (1680^2=2^8\times3^2\times5^2\times7^2). Step 3: In a perfect square, all prime exponents are even.

Step 1: (502929=3^7\times23). Step 2: Both prime factors are present in (n) with sufficient powers. Step 3: Therefore, (n) must be divisible by 502929.

Step 1: In (n), the powers are (2^{10}), (3^6), and (5^4). Step 2: (28125=3^2\times5^5), which needs power 5 of 5. Step 3: Since (n) has only (5^4), (n) is not divisible by 28125.

Step 1: Calculate (2^{11}=2048) and (3^5=243). Step 2: (2048\times243\times5=2488320). Step 3: In larger multiplication, simplify powers first.

Step 1: Calculate (2^7=128), (3^6=729), and (5^3=125). Step 2: (128\times729\times125=11664000). Step 3: Simplifying powers first keeps the calculation clear.

Step 1: In LCM, take the higher power of each prime. Step 2: The powers of 2 are 7 and 10. Step 3: The higher power is 10, so the answer is 10.

Step 1: In HCF, take the smaller power of the common prime. Step 2: The powers of 7 are 1 and 6. Step 3: The smaller power is 1, so the answer is 1.

Step 1: Write (524880=104976\times5). Step 2: (104976=16\times6561=2^4\times3^8). Step 3: Therefore, (524880=2^4\times3^8\times5).

Step 1: For a perfect cube, exponents must be multiples of 3. Step 2: (2^{12}) is already suitable, while (3^8) and (5^5) must become (3^9) and (5^6). Step 3: The smallest multiplier is (3\times5).

Step 1: (254016=2^6\times3^4\times7^2). Step 2: All powers are even, so the number is already a perfect square. Step 3: If it is already a perfect square, the smallest divisor is 1.

Step 1: In multiplication, exponents of the same prime base are added. Step 2: The power of 7 in (x) is 6 and in (y) is 7. Step 3: In (xy), the power of 7 is (6+7=13).

Step 1: Powers of the same base 2 are added in multiplication. Step 2: The power of 2 in (x) is 10 and in (y) is 9. Step 3: The total power is (10+9=19).

Step 1: Write (223092870=9699690\times23). Step 2: (9699690=2\times3\times5\times7\times11\times13\times17\times19). Step 3: Therefore, the distinct prime factors are 2, 3, 5, 7, 11, 13, 17, 19, and 23.

Step 1: The common prime factors are 2, 3, and 17. Step 2: The smaller powers are (2^6), (3^5), and (17^1). Step 3: (64\times243\times17=264384), so the HCF is 264384.

Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^8), (3^7), (5^2), and (17^2). Step 3: So the correct form is (2^8\times3^7\times5^2\times17^2).

Step 1: The product is (720\times100800). Step 2: (720=2^4\times3^2\times5) and (100800=2^6\times3^2\times5^2\times7). Step 3: The power of 2 in the product is (4+6=10).