Prime Factorisation

Step 1: The final form of prime factorisation is based only on prime numbers. Step 2: If a base like (45) remains, it must be written as (45=3^2\times5). Step 3: In exams, check every base before writing the final answer.

Step 1: Write (14256=16\times891). Step 2: (16=2^4) and (891=3^4\times11), so (14256=2^4\times3^4\times11). Step 3: Do not leave 891 in the final form.

Step 1: Write (18144=32\times567). Step 2: (32=2^5) and (567=3^4\times7), so (18144=2^5\times3^4\times7). Step 3: Give 567 its complete prime form.

Step 1: Write (24200=200\times121). Step 2: (200=2^3\times5^2) and (121=11^2), so (24200=2^3\times5^2\times11^2). Step 3: Do not leave 200 and 121 in the final form.

Step 1: Write (25410=210\times121). Step 2: (210=2\times3\times5\times7) and (121=11^2), so (25410=2\times3\times5\times7\times11^2). Step 3: Give prime form to both 210 and 121.

Step 1: Write (27216=16\times1701). Step 2: (1701=3^5\times7), so (27216=2^4\times3^5\times7). Step 3: Convert 1701 into prime form using 3 and 7.

Step 1: Write (30240=32\times945). Step 2: (945=3^3\times5\times7), so (30240=2^5\times3^3\times5\times7). Step 3: Do not keep 945 in the final answer.

Step 1: Write (36300=300\times121). Step 2: (300=2^2\times3\times5^2) and (121=11^2), so (36300=2^2\times3\times5^2\times11^2). Step 3: Convert 300 and 121 into prime powers.

Step 1: Write (38115=315\times121). Step 2: (315=3^2\times5\times7) and (121=11^2), so (38115=3^2\times5\times7\times11^2). Step 3: Break both parts completely.

Step 1: Write (43200=432\times100). Step 2: (432=2^4\times3^3) and (100=2^2\times5^2), so (43200=2^6\times3^3\times5^2). Step 3: Count the total power of 2 as 6.

Step 1: Write (48510=90\times539). Step 2: (90=2\times3^2\times5) and (539=7^2\times11), so (48510=2\times3^2\times5\times7^2\times11). Step 3: Give prime form to both 90 and 539.

Step 1: Write (52920=8\times6615). Step 2: (6615=3^3\times5\times7^2), so (52920=2^3\times3^3\times5\times7^2). Step 3: Give 6615 its complete prime form.

Step 1: Write (59290=70\times1331). Step 2: (70=2\times5\times7) and (1331=11^3), so (59290=2\times5\times7\times11^3). Step 3: Change 1331 into (11^3).

Step 1: Divide 65536 repeatedly by 2. Step 2: Sixteen factors of 2 give (65536=2^{16}). Step 3: 256 and 16 are composite bases, so write the power of 2 in final form.

Step 1: Write (68040=8\times8505). Step 2: (8505=3^5\times5\times7), so (68040=2^3\times3^5\times5\times7). Step 3: Convert 8505 into prime powers.

Step 1: Write (76230=630\times121). Step 2: (630=2\times3^2\times5\times7) and (121=11^2), so (76230=2\times3^2\times5\times7\times11^2). Step 3: Break both 630 and 121 completely.

Step 1: Write (88200=8\times11025). Step 2: (11025=3^2\times5^2\times7^2), so (88200=2^3\times3^2\times5^2\times7^2). Step 3: Convert 11025 into prime powers.

Step 1: Write (95256=8\times11907). Step 2: (11907=3^5\times7^2), so (95256=2^3\times3^5\times7^2). Step 3: Convert 11907 into powers of 3 and 7.

Step 1: Write (108900=900\times121). Step 2: (900=2^2\times3^2\times5^2) and (121=11^2), so (108900=2^2\times3^2\times5^2\times11^2). Step 3: This is a square form, so all exponents are even.

Step 1: Write (127008=32\times3969). Step 2: (3969=3^4\times7^2), so (127008=2^5\times3^4\times7^2). Step 3: Convert 3969 into prime powers.

Step 1: Write (130680=1080\times121). Step 2: (1080=2^3\times3^3\times5) and (121=11^2), so (130680=2^3\times3^3\times5\times11^2). Step 3: Break 1080 and 121 completely.

Step 1: (127008=32\times3969). Step 2: (32=2^5) and (3969=3^4\times7^2), so the power of 2 is 5. Step 3: Comparing gives (a=5).

Step 1: Write (95256=8\times11907). Step 2: (11907=3^5\times7^2), so (95256=2^3\times3^5\times7^2). Step 3: From the given form, (b=5).

Step 1: (108900=900\times121). Step 2: (900=2^2\times3^2\times5^2) and (121=11^2), so the power of 5 is 2. Step 3: Comparing gives (m=2).

Step 1: Write (68040=8\times8505). Step 2: (8505=3^5\times5\times7), so the power of 3 is 5. Step 3: Comparing gives (p=5).