Question

hello friends good morning canbyou give me answer of. Show that any positive odd integer is of the form 6q+1,or 6q+3,or 6q+5wjere q is some integer

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Answer 1

Step-by-step explanation:

To Show :-

Any positive odd integer is of the form 6q+1, 6q+3, 6q+5, where q is some integer.

Now,

Let n be the given positive odd integer.

On dividing n by 6, let q be the quotient and r be the remainder.

Then, by Euclid's division lemma, we have :

n = 6q + r, where 0 ≤ r < 6

⇝n = 6q + r, where r = 0,1,2,3,4,5

For r = 0

⇝n = 6q

For r = 1

⇝n = 6q + 1

For r = 2

⇝n = 6q + 2

For r = 3

⇝n = 6q + 3

For r = 4

⇝n = 6q + 4

For r = 5

⇝n = 6q + 5

But, n = 6q, (6q + 2), (6q + 4) give even values of n.

Thus, when n is odd, it is of the form (6q + 1) or (6q + 3) or (6q + 5) for some integer q.