Question

6. (a) In the dozenal base, ind all primes up to the dozenal number 200. Use whatever symbol you want for the digits ten and eleven (b) In the octal base, find all the primes up to the octal number 400. You just may want to use the sieve of Eratosthenes for this problem!

Answer 1

(a) Basically, in case of decimal number system, prime nos. other than (2, 3) are given by (6n$\pm$1). In case of base 12 (dozenal) system, 200 will be the decimal (2*12^2=288) number and thereby the problem can be modelled as determining all the prime nos. from 1 to 288 in base 10 (decimal) and converting it to base 12 (dozenal).

The prime nos. up-til 288 are :(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283).

In the dozenal system, we assume (a=10, b=11).

The nos. in dozenal are: (2, 3, 5, 7, b, 11, 15, 17, 1b, 25, 27, 31, 35, 37, 3b,. 45, 4b, 51, 57, 5b, 61, 67, 6b, 75, 81, 85, 87, 8b, 91, 95, a7, ab, b5, b7, 105, 107, 111, 117, 11b, 125, 12b, 131, 13b, 141, 145, 147, 157, 167, 16b, 171, 175, 17b, 181, 18b, 195, 19b, 1a5, 1a7, 1b1, 1b5, 1b7).

(b). Basically, in case of decimal number system, prime nos. other than (2, 3) are given by (6n$\pm$1). In case of base 8 (octal) system, 400 will be the decimal (4*8^2=256) number and thereby the problem can be modelled as determining all the prime nos. from 1 to 256 in base 10 (decimal) and converting it to base 8 (dozenal).

The prime nos. up-til 256 are :(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251).

The nos. in octal are: (2, 3, 5, 7, 13, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 103, 107, 111, 117, 123, 131, 141, 145, 147, 153, 155, 161, 177, 203, 211, 213, 225, 227, 235, 243, 247, 255, 263, 265, 277, 301, 305, 307, 323, 337, 343, 345, 351, 357, 361, 373).