Question

prove that square of any positive integer is of the form 5q, 5q+1,5q+4 for some integer q​

Answer 1

By Euclid division algorithm

a=bq+r

let b=5

if r=0

a=5q+0=5q

squaring both sides

(a) square =(5q)square

a square =25q square

a square =5×5q square

a square =5q

where q=5q square

if r=1

a=5q+1

squaring both sides

a square = (5q+1)whole square

a square =(5q)square+2×5q×1+(1)square

a square = 25q square +10q+1

a square =5 (5q square +2q)+1

a square =5q+1

if r=2

a=5q+2

squaring both sides

a square =(5q+2)whole square

a square =(5q)square +2×5q×2+(2)square

a square =(25q)square +20q+4

a square =5 (5q square +4q)+4

a square =5q+4