Parity of Integer equals Parity of its Square/Odd
Jump to navigation
Jump to search
Theorem
Let $p \in \Z$ be an integer.
Let $p$ be odd.
Then $p^2$ is also odd.
Proof
Let $p$ be an integer.
By the Division Theorem, there are unique integers $k$ and $r$ such that $p = 2k + r$ and $0 \le r < 2$.
That is, $r = 0$ or $r = 1$, where:
Let $r = 1$, so:
- $p = 2 k + 1$
Then:
- $p^2 = \paren {2 k + 1}^2 = 4 k^2 + 4 k + 1 = 2 \paren {2 k^2 + 2 k} + 1$
and so $p^2$ is odd.
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.14$: Exercise $18 \ (3)$