Quick solution needed

Write down explicitly all the elements in the residue classes of Z/18Z. Prove that the distinct equivalence classes in ZZ are precisely 0, 1, 2, ..., n - I ( use the Division Algorithm). Prove that if a = a, 10" + an-1 10"-1 + . . . + a1 10 + ap is any positive integer then a = an + an-1+ . .. + + 0 (mod 9) (note that this is the usual arithmetic rule that the remainder after division by 9 is the same as the sum of the decimal digits mod 9 - in particular an integer is divisible by 9 if and only if the sum of its digits is divisible by 9) [note that 10 = 1 (mod 9)]. Compute the remainder when 37100 is divided by 29. Compute the last two digits of 91500 Prove that the squares of the elements in Z/4Z are just 0 and i. Prove for any integers a and b that a2 + 62 never leaves a remainder of 3 when divided by 4 (use the previous exercise). Prove that the equation a2 + 62 = 3c2 has no solutions in nonzero integers a, b and c. [Consider the equation mod 4 as in the previous two exercises and show that a, b and c would all have to be divisible by 2. Then each of a2, b2 and c has a factor of 4 and by dividing through by 4 show that there would be a smaller set of solutions to the original equation. Iterate to reach a contradiction.] Prove that the square of any odd integer always leaves a remainder of 1 when divided by 8. Prove that the number of elements of (ZZ)* is p(n) where o denotes the Euler g