Question 2. For some integer $q$, every odd integer is of the form

Solution:

To find a general form for every odd integer in terms of some integer $$q$$, we can follow these steps:

1. Understanding Odd and Even Numbers:
– An even number can be expressed in the form $$2n$$, where $$n$$ is an integer. For example, $$0,2,4,6,\dots$$ can be represented as $$2\times 0,2\times 1,2\times 2,2\times 3,\dots$$.
– An odd number can be expressed as an even number plus one. Therefore, if $$2n$$ is an even number, then $$2n+1$$ is an odd number.

2. Defining $$q$$:
– Let $$q$$ be any integer. This means $$q$$ can take values like $$0,1,-1,2,-2,\dots$$.

3. Expressing Even Numbers:
– For any integer $$q$$, the expression $$2q$$ will yield an even number. For example:
– If $$q=0$$, then $$2q=0$$ (even).
– If $$q=1$$, then $$2q=2$$ (even).
– If $$q=2$$, then $$2q=4$$ (even).

4. Finding the General Form for Odd Integers:
– Since we know that every odd integer can be represented as an even integer plus one, we can write:
$$\text{Odd Integer}=2q+1$$
– This means that for any integer $$q$$, $$2q+1$$ will always yield an odd integer. For example:
– If $$q=0$$, then $$2\left(0\right)+1=1$$ (odd).
– If $$q=1$$, then $$2\left(1\right)+1=3$$ (odd).
– If $$q=2$$, then $$2\left(2\right)+1=5$$ (odd).

5. Conclusion:
– Therefore, we can conclude that every odd integer can be expressed in the form:
$$\text{Every odd integer}=2q+1\text{for some integer}q.$$