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Question
Show that every positive odd integer is of the form (4q + 1) or (4q + 3), where q is some integer.
Solution
According to Euclid's division lemma,
a = bq + r where 0 ≤ r ≤ b
Now, let a be an odd positive integer and b = 4.
When 0 ≤ r ≤ 4 so, the possible values of r will be 0, 1, 2, 3.
Now, the possible values of a will be thus, 4q, 4q + 1, 4q+2, 4q +3 where q is an integer.
But, we already know that a is an odd positive integer.
So, a will be 4q + 1 and 4q + 3.
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