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Question:

Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

Answer:

Suppose a be any positive integer and b = 6. Then, by Euclid’s algorithm, ๐ = ๐๐ + ๐, for some integer ๐ ≥ ๐, and ๐ = ๐,๐,๐,๐,๐,๐, because of ๐≤๐<๐.

Now substituting the value of r, we get,

If r = 0, then ๐ = ๐๐

Similarly, for ๐= ๐,๐,๐,๐ and 5, the value of a is ๐๐+๐,๐๐+๐,๐๐+๐,๐๐+๐ and ๐๐+๐, respectively.

If ๐ = ๐๐,๐=๐๐+๐ ๐๐ง๐ ๐ = ๐๐+๐, then a is an even number and divisible by 2.

A positive integer can be either even or odd. Therefore, any positive odd integer is of the form of ๐๐+๐,๐๐+๐ ๐๐ง๐ ๐๐+๐, where q is some integer.

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