Question

Show that square of any positive integer cannot be of the form of 5m + 2 or 5m + 3 for some integer m.
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Answer 1

Answer:let the positive let the positive integer be a

a=bq+r

b=5                    r=0,1,2,3,4

therefore,

a=5q

a=5q+1

a=5q+2

a=5q=3

a=5q+4

Case1: a=5q

square both sides

a²=(5q)²

a²=25q²

a²=5(5q²)

a²=5m

m=(5q²)

case 2: a=5q+1

square both sides

a²=(5q+1)²

a²=25q²+1+10q

a²=5(5q²+2q)+1

a²=5m+1

m=(5q²+2q)

Case3: a=5q+2

square both sides

a²=(5q+2)²

a²=25q²+4+20q

a²=5(5q²+4q)+4

a²=5m+4

m=(5q²+4q)

Case4: a=5q+3

square both sides

a²=(5q+3)²

a²=25q²+9+30q

a²=25q²+5+4+30q

a²=5(5q²+1+6q)+4

a²=5m+4

m=(5q²+1+6q)

Case 5:a=5q+4

square both sides

a²=(5q+4)²

a²=25q²+16+40q

a²=25q²+15+1+40q

a²=5(5q²+3+8q)+1

a²=5m+1

m=(5q²+3+8q)

therfore, the square of any positive integer cannot be of the form 5m+2 or 5m+3 for some integer m but can be of the form 5m, 5m+1 or 5m+4.