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 1

SOLUTION :  

Let ‘a’ be any odd positive integer and b = 6.

Then, there exists integers q and r such that

a = 6q + r, 0 ≤ r < 6 , [r = 0,1,2,3,4,5]

[By using  division algorithm]

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

But , 6q or 6q + 2 or 6q + 4 are even positive integers.

So, a = 6q + 1 or 6q + 3 or 6q + 5 are odd positive integers.

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

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Answer 2

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