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

Let a be a given integer.

By Euclid's Division Lemma

a = 6q + r

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

When r = 0

$a = 6q \\a = 2(2q)$

Since it is divisible by 2

it is an even number.

When r = 1

$a = 6q + 1$

Odd number

When r = 2

$a = 6q + 2\\a = 2(2q+1)$

Since it is divisible by 2

it is an even number.

When r = 3

$a=6q + 3$

Odd number

When r = 4

$a=6q + 4\\a=2(3q+2)$

Since it is divisible by 2

it is an even number.

When r = 5

$a= 6q + 5$

Odd number

$\boxed{\boxed{\bold{Therefore, \ any \ positive \ odd \ integer \ is \ of \ the \ form \ 6q+1, \6q+3 \ or \ 6q+5. }}}}}}}$

                                                             

Answer 2

Step-by-step explanation:

from Euclid division Lemma, we have

a=bq+r; 0<r<b, let as assume b=6; 0_<r<6,where r=1,2,3,4 and 5.

when i) r=0, a=6q+0,a=6q( can even integer)

ii) r=1,a=6q+1,( can odd integer)

iii) r=2 ,a=6q+2,(can even integer)

iv) r=3,a=6q+3,(can odd integer)

v) r=4,a=6q+4,(can even integer)

vi) r=5,a=6q+5,(can odd integer)

as,"a" is odd,

therefore, any positive odd integer is of the form 6q+1,6q+3, and 6q+5...