Question

show that the square of any positive integer cant be of the form 5m+2or 5m+3 where m is a whole number​

Answer 1

Step-by-step explanation:

Let a and b be positive integers

a=bq+r

0≤r<b

According to the question

Let b=5 r=0,1,2,3,4

a=5q+r

Case 1:- r=0

a=5q

a²=(5q)²

a²=25q²

a²=5(5q²)

Let 5q²=m

a²=5m

Case 2:- r=1

a=5q+1

a²=(5q+1)²

a²=25q²+10q+1

a²=5(5q²+2q)+1

Let 5q²+2q be m

a²=5m+1

Case 3:- r=2

a=5q+2

a²=(5q+2)²

a²=25q²+20q+4

a²=5(5q²+4q)+4

Let 5q²+4q be m

a²=5m+4

Case 4:- r=3

a=5q+3

a²=(5q+3)²

a²=25q²+30q+9

a²=25q²+30q+5+4

a²=5(5q²+6q+1)+4

Let 5q²+6q+1 be m

a²=5m+4

Case 5:- r=4

a=5q+4

a²=(5q+4)²

a²=25q²+40q+16

a²=25q²+40q+15+1

a²=5(5q²+8q+3)+1

Let 5q²+8q+3 be m

a²=5m+1

The square of any integer can be in the form 5m+1 and 5m+4

But it can't be in the form 5m+2 and 5m+3

Hence proved