Question

Show that any positive odd integer is of the form 4q+1or 4q+3,where q is a positive integer

Answer 1

Hey

Here is your answer,

Let a be the given positive odd integer.
On dividing a by 4,let q be the quotient and r the remainder.
Therefore,by Euclid's algorithm we have
a = 4q + r 0 ≤ r < 4
⇒ a = 4q + r r​ = 0,1,2,3
⇒ a = 4q, a = 4q + 1, a = 4q + 2, a = 4q + 3
But, 4q and 4q + 2 = 2 (2q + 1) = even
Thus, when a is odd, it is of the form (4q + 1) or (4q + 3) for some integer q.

Hope it helps you!

Answer 2

$\huge \bf{ \red{hey}}$

$\huge{ \mathfrak{ \blue{here \: is \: answer}}}$

let a be any positive integer

then

b= 4

a= bq+r

0≤r<b

0≤r<4

r= 0,1,2,3

case 1.

r=0

a= bq+r

4q+0

4q

case 2.

r=1

a= 4q+1

6q+1

case3.

r=2

a=4q+2

case 4.

r=3

a=4q+3

hence from above it is proved that any positive integer is of the form 4q, 4q+1,4q+2,4q+3

$\huge \boxed{ \boxed{ \green{HOPE\: IT \: HELPS}}}$

$\huge{ \pink{thanks}}$