prove that any odd positive integer of
8q+1,where q is any integer?
To prove that any odd positive integer can be written in the form of 8q + 1, where q is any integer, we can use mathematical induction.
First, let's verify that the statement is true for a specific value of q, say q = 0.
When q = 0, 8q + 1 = 8(0) + 1 = 1. Since 1 is an odd positive integer, the statement holds true for q = 0.
Now, let's assume that the statement is true for some value of q = k, where k is any positive integer. In other words, assume that 8k + 1 is an odd positive integer.
Using this assumption, let's prove that the statement is also true for q = k + 1.
When q = k + 1, 8q + 1 = 8(k + 1) + 1 = 8k + 8 + 1 = 8k + 9.
To verify that 8k + 9 is an odd positive integer, we can rewrite it as (8k + 8) + 1 = 8(k + 1) + 1.
We can clearly see that 8(k + 1) + 1 is of the form 8q + 1, where q = k + 1. Hence, the statement holds true for q = k + 1.
By using the principle of mathematical induction, we have proved that the statement is true for q = 0, and if it is true for q = k, then it is true for q = k + 1. Therefore, the statement is true for all odd positive integers.