IMPORTANT NOTE: This Answer was posted already but when discussing with the instuctor, it was found to be INCORECT. SO PLEASE dont post the answer that have been already posted online. Please show the step for 1a to 1f individually.

1. (30 points) For this problem, consider your student identification number as the integer n. You can use on-line programs (or write your own) to answer the questions, but, if the directions for a problem include the statement "Justify your answer" then you must show intermediate values in your calculations and not just a final answer. For example you can answer 1a and 1b with just the correct prime numbers, but you must answer 1d with intermediate calculations 1a. Find the prime factorization of n. 1b. If 3k E Z|n-2k, find prime pi,prime p2 such that n- pi + p2 else find prime pi,prime p2 such that n+1-pi + p2. (According to Goldbach's Conjecture the primes exist.) 1c. Apply the quotient remainder theorem to n and d = 52 to find unique q and r. 1d. Is the following proposition true or false? Justify your answer. n mod 1e. Calculate 103 * 103 10 1f. Generate the value an (n is your student ID) based on the following definition, n- 0 10 Justify your conclusion. Explain what an represents about the digits of n and for full credit explain why the definition gives that result (in particular, what purpose does ajn| serve?) 10 1. (30 points) For this problem, consider your student identification number as the integer n. You can use on-line programs (or write your own) to answer the questions, but, if the directions for a problem include the statement "Justify your answer" then you must show intermediate values in your calculations and not just a final answer. For example you can answer 1a and 1b with just the correct prime numbers, but you must answer 1d with intermediate calculations 1a. Find the prime factorization of n. 1b. If 3k E Z|n-2k, find prime pi,prime p2 such that n- pi + p2 else find prime pi,prime p2 such that n+1-pi + p2. (According to Goldbach's Conjecture the primes exist.) 1c. Apply the quotient remainder theorem to n and d = 52 to find unique q and r. 1d. Is the following proposition true or false? Justify your answer. n mod 1e. Calculate 103 * 103 10 1f. Generate the value an (n is your student ID) based on the following definition, n- 0 10 Justify your conclusion. Explain what an represents about the digits of n and for full credit explain why the definition gives that result (in particular, what purpose does ajn| serve?) 10