Question

Prove that the square of any positive integer is of the form 5q , 5q+1 , 5q + 4 for some integer q. I need brainliest answer friends.​

Answer 1

$\begin{gathered}{\Huge{\textsf{\textbf{\underline{\underline{\purple{Answer:}}}}}}}\end{gathered}$

$\implies$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 three cases x2 is either of the form 5q or 5q+1 or 5q+4 and for integer q.

• I Hope it's Helpful My Friend.

Answer 2

Step-by-step explanation:

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