MATH 124 Midterm: MATH 124 Harvard Midterm124 Fall 03
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Collaboration: on the midterm no collaboration is allowed. Show all your working, and write up your solutions as neatly as possible. The points on the exam total 105, although only the rst 100 points will count. If you have any questions, don"t hesitate to email me at fcale@math. harvard. edu: (20 points) find the general solution to the following simultaneous equations: x 55 mod 77, x 33 mod 99, x 99 mod 111. Show all your working: (10 points) state the law of quadratic reciprocity, and use it to calculate which of the primes p = 2, 3, 5 and 7 are quadratic residues modulo 2003. You may assume that 2003 is prime: (10 points) let p 5 be prime. Show that the sum of all the quadratic residues is 0 modulo p. Hint: show that if m is composite, then (m) < m 1. (c) (7 points) let k 1, and let m = 22k.
