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Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer. |
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Answer» By Euclid’s division algorithm, a = bq + r Take b = 4 ∴ Since 0 ≤ r < 4,r = 0, 1, 2, 3 So, a = 4q, 4q + 1, 4q + 2, 4q + 3 Clearly, a = 4q, 4q + 2 are even, as they are divisible by 2. Therefore 'a' cannot be 4q, 4q + 2 as a is odd. But 4q + 1, 4q + 3 are odd, as they are not divisible by 2. ∴ Any positive odd integer is of the form (4q + 1) or (4q + 3). |
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