Quotient-Remainder Theorem; Proofs Involving Cases
Tom Kelliher, MA 115
Oct. 17, 1997
Final: 9:00--11:00am on 12/15/97 in HS 149.
Read Section 3.6. Problems due Wed. 10/22: 3.3.31, 3.3.33, 3.3.38,
3.4.20, and 3.4.31.
For any integer n and any positive integer d, there exist
unique integers q and r such that
Examples:
- n=12, d=5.
- n=37, d=4.
- n=-39, d=13.
- Div: is the quotient of n divided by d.
- Mod: is the remainder of n divided by d.
If :
- What is ?
- What is ?
- What C++ operators correspond to div and mod?
Prove that any two consecutive integers have opposite parity.
- What's parity?
- This is an example of the proof by division into cases argument form
(Example 1.3.8).
Prove that the square of any odd integer can be written as 8m+1 for some
integer m.
- Try it using only the definition of odd.
- Show that any integer can be represented modulo 4.
- Try it using the modulo 4 representation of integers.
Prove that the product of any two consecutive integers is even. Two forks
in the path:
- Prove using the fact that any two consecutive integers have opposite
parity.
- Prove by representing integers modulo 2.
Thomas P. Kelliher
Thu Oct 16 11:51:44 EDT 1997
Tom Kelliher