(ii) The first few cubes are 1,8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, and 1728. Write 1729 as a sum of two cubes in two different ways and hence factorise 1729 as a product of primes. Show that if p is a prime factor of 1729, and a is a number with (a, p) = 1, then a 1728 = 1 mod p, and hence if (a, 1729) = 1, then 1728 = 1 mod 1720

Consider the list of numbers: 2n-1, where n first equals 2, then 3, 4, 5,6,…What is the smallest value of n for which 2n -1 is not a prime number.  Show all your work.

Answer the following questions where div is for finding integer quotient and mod is for remainder. 30 div 4 = -23 div 9 = 27 div 4 = -24 div 7 = 21 div 7 = 30 mod 4 = -23 mod 9 = 27 mod 4 = -24 mod 7 = 21 mod 7 = -23 div 8 = -23 mod 8 = [ 174465950 mod 101 = 788751760 mod 101 =

4. Determines the result of the following operations: a. (27 – 3) (mod 5) b. (56 – 14) (mod 7) c. (6+7) (mod 2)