Question

show that square of any positive integer is of the form 5q, 5q+1 or 5q+4
for the same integer q

Answer 1

Let x be any positive integer

Then x = 5q or x = 5q+1 or x = 5q+4  for integer x.

If x = 5q, x2 = (5q)2 = 25q2 = 5(5q2) = 5n (where n = 5q2 )

If x = 5q+1, x2 = (5q+1)2 = 25q2+10q+1 = 5(5q2+2q)+1 = 5n+1 (where n = 5q2+2q )

If x = 5q+4, x2 = (5q+4)2 = 25q2+40q+16 = 5(5q2 + 8q + 3)+ 1 = 5n+1 (where n = 5q2+8q+3 )

∴in each of three cases x2 is either of the form 5q or 5q+1 or 5q+4 and for integer q.

Answer 2

$\huge \bf{ \red{hey}}$

$\huge{ \mathfrak{here \: is \: answer}}$

let a be any positive integer

then

b=5

0≤r<b

0≤r<5

r=0,1,2, 3,4

case 1.

r=0

a=bq+r

5q+0

5q

case 2.
r=1
a=bq+r

5q+1

case 3.

r=2

5q+2

case4.

r=3
5q+3

case 5.

r=4

5q+4

from above it is proved.

$\huge \boxed{ \boxed{ \pink{hope \: it \: helps}}}$

$\huge{ \green{thanks}}$