Even and odd integers; divisibility

foundations

Published

October 11, 2026

To do

Read §4 Even and odd integers; divisibility
Before Theorem 1, Theorem 2, Corollary: cover the proof, try it yourself
§4 exercises: ★1–★7, 8, 10, 12, 14, 16, 18, 20, 22, ★24, ★25, ★26
Re-do diagnostic problem 26 (odd + odd) cold. Compare with your first attempt
Paper: all listed exercises. Mark ✓ / ✗ / S. Answers: back of the book.
Notebook: write each div below in your own words. Copy the ★ exercises in clean.

Definition 1 (Even (sudé) and odd (liché) integer) Your words.

Theorem 1 (Theorem 1: parity (parita) of a sum) Your words.

Proof. Cover the book’s proof. Try 10–15 min first.

Theorem 2 (Theorem 2: parity of a square) Your words.

Proof. Cover the book’s proof. Try 10–15 min first.

Theorem 3 (Corollary (důsledek): from a^2 back to a) Your words.

Proof. Cover the book’s proof. Try 10–15 min first.

Definition 2 (Divisibility (dělitelnost)) Your words.

Intuition

Your words.

Exercises

★ Diagnostic 26. Prove: the sum of two odd integers is even.

Solution

Your words.

★ 1. Prove the cases of Theorem 1 that the text does not prove.

Solution

Your words.

★ 2. Prove: if a is even and b is any positive integer, then ab is even.

Solution

Your words.

★ 3. Prove: if a is even, then a^3 is even.

Solution

Your words.

★ 4. Prove: if a is odd, then a^3 is odd.

Solution

Your words.

★ 5. Prove: if n is even, then (-1)^n = 1.

Solution

Your words.

★ 6. Prove: if n is odd, then (-1)^n = -1.

Solution

Your words.

★ 7. Prove: if m, n are odd, then mn is odd.

Solution

Your words.

★ 24. Prove: if a \equiv b \pmod 5 and x \equiv y \pmod 5, then a + x \equiv b + y \pmod 5 and ax \equiv by \pmod 5.

Solution

Your words.

★ 25. Same as 24, for any positive integer d in place of 5.

Solution

Your words.

★ 26. Assume every positive integer has the form 3k, 3k + 1 or 3k + 2. Prove: if 3 \mid n^2, then 3 \mid n.

Solution

Your words.