Math 5336 Number Theory Dr. Cordero
Class activities for October 14th, 2002
Topics: Primes and Their Distribution, II
The Theory of Congruences, I
· Page 44 # 3 (a, c, e ), 4, 5a (Hint: Use Corollary 1, p. 41), 6a (Hint: Write and make “obvious” choices for the coefficients.)
· Page 59 # 3, 9
i. How are the numbers within a specific list related?
ii. How are the numbers in different lists related?
iii. How many distinct lists are there?
7.Conjecture: Let a, b c, d and m be integers with m>0. Assume that
· a is congruent to b modulo m
· c is congruent to d modulo m
Then we have;
· a+c is congruent to b+d modulo m
· ac is congruent to bd modulo m
Prove this conjecture. (Hint: You’ve already done this!)
8. Definition: If m>0 and r is the remainder when the division algorithm is used to divide b by m, then r is called the least residue of b modulo m.
9. Practice: Find the least residue:
· 93 modulo 17
· 421 modulo 17
· 93 + 421 modulo 17
· (93)(421) modulo 17
· modulo 21.
10. General method to find the least residue of modulo m:
Step 1: Write z as a sum of powers of 2.
Step 2: Successively square a until you’ve gone as high as you need, reducing modulo m at each stage. Feel free to use negative numbers if it makes the computations easier.
Step 3: Put it together, using laws of exponents.
11. Compute the least residue of modulo 17.
12. At-home Practice: Find the least residue of modulo 4; modulo 19; modulo 23
13. Find the last two digits of .
14. Homework: Pp. 68-69 # 2, 4, 5, 16.