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Real Numbers

Question 2:

Show that any positive odd integer is of the form , or , or , where q is some integer.

Answer:

Let a be any positive integer and b = 6. Then, by Euclid’s algorithm,

a = 6q + rfor some integer q0, and r = 0, 1, 2, 3, 4, 5 because 0 r < 6.

Therefore, a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5

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