Math 58 - Number Theory

September 19, 2006

Homework 5 (due September 21):

Read Section 4.1, and be absolutely confident that you can do problems like ## 1-7 and 12-15. Also ## 8-10, for that matter; but you don’t need to turn any of those in.

For 9/21, read Section 4.2; the main point is that certain numbers have multiplicative inverses if we work mod m. If you like, peek at 4.3.

1. Find the least non-negative residues of…

4! (mod 5)

5! (mod 6)

6! (mod 7)

7! (mod 8)

8! (mod 9)

12! (mod 13)

…and guess the least non-negative residue of

19080! (mod 19081)

2. Calculate 2¹³ (mod 13), without ever using a 4-digit number.

3. Calculate:

2¹⁷²⁹ (mod 1729)

3¹⁷²⁹ (mod 1729)

Note: 1729 = 12³ + 1³ = 10³ + 9³ = 1 + 2⁶ + 2⁷ + 2⁹ + 2¹⁰.

4. Is 1729 prime ?

5. Show that if a is any odd integer, then a² = 1 (mod 8).

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