Prove that if n is an odd positive integer, then 1 ≡ n(power of 2) (mod 4).
To prove that if n is an odd positive integer, then 1 ≡ n^2 (mod 4), we can use modular arithmetic.
First, let's consider any odd positive integer n. Since n is odd, we can write it as n = 2k + 1, where k is a non-negative integer.
Now, let's compute n^2 (mod 4):
n^2 = (2k + 1)^2 = 4k^2 + 4k + 1
When we divide n^2 by 4, we get a quotient of k^2 + k and a remainder of 1. So, n^2 = 4k^2 + 4k + 1 ≡ 1 (mod 4).
Therefore, we can conclude that if n is an odd positive integer, then 1 ≡ n^2 (mod 4).
Use Mathematical Induction to prove that 1
1
.2 1 2 2.
n
i n
r
i n