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

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

1.	show that any positive odd integer is of the form 4q+1 or 4q+3,where a is a positive integer.
Answer» By Euclid\'s division algorithm,a = bq + r = 4q + rTake b = 4.Since, 0 {tex}\\leqslant{/tex}\xa0r < 4, r = 0,1, 2, 3{tex} a=4q,4q+1,4q+2 ,4q+3{/tex}Clearly, a =4q=2(2q) and\xa04q+2=2×(2q+1)So 4q and 4q+2 are evenTherefore 4q + 1, 4q + 3 are odd, as they are proceeding numbers of even numbers 4q and 4q+2.{tex}\\therefore{/tex}\xa0Any positive odd integer is of form 4q+1 or 4q+3 .Where q is a positive integer.