Question

show that any positive integer is of the form 4q 1 or 4q 3 where q is some integer

Answer 1

the question is not rightly  formed.... It should be any odd positive integer.. Also  q should be a positive integer of 0.

any integer when divided by 4 gives a quotient q, and a reminder which is either 0, 1, or 2 or 3...  

so N = 4q    or  4 q +1  or  4 q+2  or  4 q+3      for a nonnegative integer q.
       4 q and  4 q + 2  are  even numbers. it is quite obvious.

  hence any odd positive integer is of the form  4q +1  or  4 q +3

equivalently also we can say  N is of the form:  4 q - 1  or  4q - 3

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

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

$\huge{ \pink{thanks}}$