Question

show that any positive odd integer is form 6q+1 or 6q+3 or 6q+5 where q is some integer​

Answer 1

Answer:

I use p instead of q.

Let a be any arbitrary positive integer.

On dividing a by 6 we get,

a=6p+r where r >6

Case 1 -Let r=O

a=6p(a is even)

Case 2-Let r= 1

a=6p+1

Case 3-Let r=2

a=6p+2(a is even)

Case 4-Let r=3

a=6p +3

Case 5-Let r=4

a=6p+4(a is even)

Case 6-Let r=5

a=6p+5

Hence,every odd integer can be written as (6p+1),

(6p+3) and (6p+5).

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Answer 2

Answer:

$\displaystyle\huge\red{\underline{\underline{AnSWEr}}}$

Let 'a' be any positive integer and b=6

Apply Euclid division lemma to A and B

$a = 6q + r \: \: \: \: where \: 0 \leqslant r < 6$

r=0,1,2,3,4,5

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

and. a is positive odd integer

a≠6q. or a≠ 6q+2 or a≠6q+4

And. a=6q+1 ,a=6q+3 , a=6q+5

Hence proved