Fundamental Theorem of Arithmetic

Step 1: Write (14700=147\times100). Step 2: (147=3\times7^2) and (100=2^2\times5^2), so (14700=2^2\times3\times5^2\times7^2). Step 3: 147 and 100 are composite, so do not keep them in the final form.

Step 1: (1485=3^3\times5\times11). Step 2: All these factors are present in the prime factorisation of (n). Step 3: Therefore, (n) must be divisible by 1485.

Step 1: In (n), the powers are (2^4), (3^2), and (5^1). Step 2: (75=3\times5^2), which needs power 2 of 5. Step 3: Since (n) has only (5^1), (n) will not be divisible by 75.

Step 1: Calculate (2^7=128) and (3^2=9). Step 2: (128\times9\times5=5760). Step 3: In larger products, simplifying powers first is safer.

Step 1: Calculate (2^4=16), (3^5=243), and (5^2=25). Step 2: (16\times243\times25=97200). Step 3: Simplifying powers first makes the calculation easier.

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

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

Step 1: (15876=126^2). Step 2: Since (126=2\times3^2\times7), (126^2=2^2\times3^4\times7^2). Step 3: In a perfect square, every prime exponent is even.

Step 1: For a perfect cube, exponents must be multiples of 3. Step 2: We must make (2^7), (3^5), and (5^4) into powers 9, 6, and 6. Step 3: So the smallest multiplier is (2^2\times3\times5^2).

Step 1: (6615=3^3\times5\times7^2). Step 2: For a perfect square, all exponents must be even, but powers of 3 and 5 are odd. Step 3: Dividing by (3\times5=15) leaves (3^2\times7^2), a perfect square.

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

Step 1: Powers of the same base 2 are added in multiplication. Step 2: The power of 2 in (x) is 6 and in (y) is 4. Step 3: The total power is (6+4=10).

Step 1: Write (30030=2310\times13). Step 2: (2310=2\times3\times5\times7\times11). Step 3: Therefore, the distinct prime factors are 2, 3, 5, 7, 11, and 13.

Step 1: The common prime factors are 2, 3, and 13. Step 2: The smaller powers are (2^2), (3^1), and (13^1). Step 3: (4\times3\times13=156), so the HCF is 156.

Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^4), (3^3), (5), and (13^2). Step 3: (16\times27\times5\times169=365040), so the answer is 365040.

Step 1: The product is (72\times1800). Step 2: (72=2^3\times3^2) and (1800=2^3\times3^2\times5^2). Step 3: The power of 3 in the product is (2+2=4).

Step 1: The product is (42\times30030). Step 2: 42 has no factor 13 and (30030=2\times3\times5\times7\times11\times13). Step 3: Therefore, the power of 13 in the product is (0+1=1).

Step 1: (14641=121\times121). Step 2: Since (121=11^2), (14641=11^4). Step 3: 121 is composite, so (121^2) is not the final prime factorisation.

Step 1: In a perfect cube, every exponent must be a multiple of 3. Step 2: Reduce (2^5) to (2^3) by dividing by (2^2), and reduce (3^4) to (3^3) by dividing by 3. Step 3: So the smallest divisor is (2^2\times3).

Step 1: First calculate (2^4=16) and (3^3=27). Step 2: (16\times27\times5\times7=15120). 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 (2^4\times3=48). Step 2: LCM is (2^7\times3^3=3456). Step 3: The ratio is (3456\div48=72).

Step 1: In (xy), exponents of the same bases are added. Step 2: Powers become (2^8), (3^6), and (5^5). Step 3: Counting with repetition gives (8+6+5=19).

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

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