We need to show that 4 divides 1-n2 whenever n is an odd positive integer.
If n is an odd positive integer then by definition
n = 2k+1 for some non negative integer, k.
Now 1 - n2 = 1 - (2k+1)2 = -4k2-4k = 4 (-k2-4k).
k is a nonnegative integer, hence -k2-4k is an integer. Thus by definition
of divisibility we conclude that 4 divides 1-n2.
To show that 4 divides 1 - n^2 whenever n is an odd positive integer, we can use the definition of odd numbers and the properties of divisibility.
Let's start by assuming that n is an odd positive integer. By definition, an odd number can be represented as 2k + 1, where k is a non-negative integer.
Substituting n in the expression 1 - n^2, we get:
1 - (2k + 1)^2 = 1 - (4k^2 + 4k + 1) = 1 - 4k^2 - 4k - 1 = -4k^2 - 4k
Factoring out 4 from the expression, we have:
-4k^2 - 4k = 4(-k^2 - k)
Now, let's take a closer look at -k^2 - k. Since k is a non-negative integer, both -k^2 and -k are integers.
Hence, -k^2 - k is also an integer. Therefore, by the definition of divisibility, we conclude that 4 divides 1 - n^2 whenever n is an odd positive integer.