Question
Show that every positive odd integer is of the form (4q + 1) or (4q + 3) for some integer q.
✓
Answer
Let 'a' be a given positive od integer.
On dividing 'a' by 4, let q be the quotient and r be the remainder.
Then, by euclid's algorithm, we have
a = 4q + r, where 0 ≤ r < 4
⇒ a = 4q + r, where r = 0, 1, 2, 3
⇒ a = 4q or a = 4q + 1 or a = 4q + 2 or a = 4q + 3
But, a = 4q and a = 4q + 2 = 2(2q + 1) are clearly even.
Thus, when 'a' is odd, it is of the form:
a = (4q + 1) or (4q + 3) for some integer q.
On dividing 'a' by 4, let q be the quotient and r be the remainder.
Then, by euclid's algorithm, we have
a = 4q + r, where 0 ≤ r < 4
⇒ a = 4q + r, where r = 0, 1, 2, 3
⇒ a = 4q or a = 4q + 1 or a = 4q + 2 or a = 4q + 3
But, a = 4q and a = 4q + 2 = 2(2q + 1) are clearly even.
Thus, when 'a' is odd, it is of the form:
a = (4q + 1) or (4q + 3) for some integer q.
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