Question

show that square of any positive integer is of 4q, 4q + 1. q is any positive intizer 4m or 4m+1

Answer 1

if a is positive integer then according to the euclide division lema
a = bq + r    (0≥r >b )

take b = 2


a = 2q + r    (0≥r>2)

so, values of r is

r ⇒ 0,1

1) if r = 0 then,

  a = 2q+0

  a = 2q

  if a is square number then

 a = (2q)²
 
 a = 4q²
 
 a = 4(q²)

 a = 4m    (here m is any positive integer and m = q²)

 
 2) if r = 1 then
 
  a = 2q + r
 
  a = 2q + 1

  if a is square number then,

  a = (2q + 1 )²
 
  a = 4q² + 4q +1

  a = 4(q² + q) +1

  a = 4m + 1     (here m is positive integer and m = q² + q )