Question

pls solve this. ( show that the square of any positive integer is of the form 5m,5m+1,5m+4 for some integer m.)

Answer 1

let us suppose that the m is positive integer 0,1,2,3,4, (
$0 < m > 5$

then, put the m=0
5m+0=5m

5m+1, 5m+2,5m+3 , 5 m +4
then m is positive integers

Answer 2

Hey!

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Let a be any integer in form of 5q, 5q+1 , 5q+2

a= 5q

•Squaring both sides :-

(a)² = (5q)²

a² = 25q²

a² = 5q(5q)

•Let m = q(5q)

•a² = 5m..... Proved

> Now second :-

a = 5q+1

Squarring both sides :-

a² = (5q+1)²

a² = (5q)² + (1)² + 2.5q.1

a² = 25q² + 1 + 10q

a² = 5(5q² + 2q) +1

•Let m = 5q²+2q)

a²= 5m + 1

Third case :-

5q + 2

a² = (5q+2)²

a² = (5q)² + 2.5q.2 + (2)²

a² = 25q² + 20q + 4

a² = 5(5q² + 4q) + 4

•Let m = 5q²+4q

a² = 5m + 4 ...... Proved

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Regards :)

Yash Raj ✌