Question

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

Answer 1

Any positive odd integer is of the form 4q+1 or 4q+3. show that any positive odd integers is of the form 4 q + 1 or 4 q + 3 where q is a positive integer. We know by Euclid's algorithm, if a and b are two positive integers, there exist unique integers q and r satisfying, where .

Answer 2

from Euclid's division algorithm we know a = bq + r ,where 0 ≤r < b
here b = 4 so r can be = 0,1,2,or 3 and q be any positive integer

 when r = 0 
a= 4q              (it is divisible by two. so it is an even number)    

 when r = 1
a = 4q + 1       ( it is an odd number)

      when r = 2
a = 4q +2       ( it is divisible by two. so it is an even number)

      when r = 3
a = 4q + 3        ( it is an odd number)
 
    From the above result it is clear that any positive odd integer is of the form 4q+1 or 4q+3 where q is a positive integer