Fundamental Theorem of Arithmetic

Step 1: In HCF, take the smaller power of the common prime. Step 2: The powers of 7 are 1 and 5. Step 3: The smaller power is 1, so the answer is 1.

Step 1: Write (262440=52488\times5). Step 2: (52488=8\times6561=2^3\times3^8). Step 3: Therefore, (262440=2^3\times3^8\times5).

Step 1: For a perfect cube, exponents must be multiples of 3. Step 2: Make (2^{11}), (3^7), and (5^4) into powers 12, 9, and 6. Step 3: The smallest multiplier is (2\times3^2\times5^2).

Step 1: (254016=64\times3969=2^6\times3^4\times7^2). Step 2: All exponents 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 5 and in (y) is 6. Step 3: In (xy), the power of 7 is (5+6=11).

Step 1: Powers of the same base 2 are added in multiplication. Step 2: The power of 2 in (x) is 9 and in (y) is 8. Step 3: The total power is (9+8=17).

Step 1: Write (9699690=510510\times19). Step 2: (510510=2\times3\times5\times7\times11\times13\times17). Step 3: Therefore, the distinct prime factors are 2, 3, 5, 7, 11, 13, 17, and 19.

Step 1: The common prime factors are 2, 3, and 13. Step 2: The smaller powers are (2^5), (3^4), and (13^1). Step 3: (32\times81\times13=33696), so the correct HCF is 33696.

Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^7), (3^6), (5^2), and (13^2). Step 3: (128\times729\times25\times169=394221600), so the correct value is 394221600.

Step 1: The product is (360\times50400). Step 2: (360=2^3\times3^2\times5) and (50400=2^5\times3^2\times5^2\times7). Step 3: The power of 2 in the product is (3+5=8).

Step 1: The product is (143\times510510). Step 2: (143=11\times13), and 510510 has 13 to power 1. Step 3: Therefore, the power of 13 in the product is (1+1=2).

Step 1: Write (160000=16\times10000). Step 2: (16=2^4) and (10000=10^4=2^4\times5^4). Step 3: Therefore, (160000=2^8\times5^4).

Step 1: In a perfect cube, every exponent must be a multiple of 3. Step 2: Reduce (2^{11}) to (2^9), (3^{10}) to (3^9), and (5^7) to (5^6). Step 3: So the smallest divisor is (2^2\times3\times5).

Step 1: Calculate (2^7=128) and (3^4=81). Step 2: (128\times81\times5\times7=362880). Step 3: To get the number from prime factorisation, multiply all factors.

Step 1: For two numbers, HCF (\times) LCM equals the product of the two numbers. Step 2: In prime powers, the smaller and higher exponents together give the total exponent. Step 3: Therefore, the answer is (ab).

Step 1: HCF is (2^7\times3^3). Step 2: LCM is (2^{10}\times3^6). Step 3: The ratio is (2^3\times3^3=8\times27=216).

Step 1: In (xy), exponents of the same bases are added. Step 2: Powers become (2^{14}), (3^{11}), and (5^{11}). Step 3: Counting with repetition gives (14+11+11=36).

Step 1: In LCM, take the higher power. Step 2: The powers of 5 are 4 and 7. Step 3: The higher power is 7, so the answer is 7.

Step 1: In HCF, take the smaller power. Step 2: The powers of 3 are 5 and 8. Step 3: The smaller power is 5, so the answer is 5.

Step 1: The theorem says that prime factorisation of a number greater than 1 is fixed. Step 2: The order may change, but the prime bases and their powers do not change. Step 3: Keep only prime factors in the final answer.