Show that any positive odd integer is of the form $4q+1$ or $4q+3,$ where $q$ is some integer.

(N/A) Let $a$ be any positive odd integer. According to Euclid's division algorithm,for any two positive integers $a$ and $b$,there exist unique integers $q$ and $r$ such that $a = bq + r$,where $0 \leq r < b$.
Here,we take $b = 4$. Thus,$a = 4q + r$,where $0 \leq r < 4$.
The possible values for the remainder $r$ are $0, 1, 2,$ and $3$.
This means $a$ can be expressed as $4q, 4q+1, 4q+2,$ or $4q+3$.
Since $a$ is an odd integer,it cannot be divisible by $2$.
- $4q = 2(2q)$,which is divisible by $2$ (even).
- $4q+2 = 2(2q+1)$,which is divisible by $2$ (even).
Therefore,$a$ cannot be $4q$ or $4q+2$.
Hence,any positive odd integer must be of the form $4q+1$ or $4q+3$.