State Euclids division lemma. Using it show that square of any positive integer is either of the form 5m or 5m+-1, where m is an integer.

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Let x be any integer

Then,

Either x=5m or x=5m+1 or x=5m+2 or, x=5m+3 or x=5m+4 for integer x. [ Using division algorithm]

If x=5m

On squaring both side and we get,

x

2

=25m

2

=5(5m

2

)=5n where n=5m

2

If x=5m+1

On squaring both side and we get,

x

2

=(5m+1)

2

=25m

2

+1+10m

=5(5m

2

+2m)+1(where5m

2

+2m=n)

=5n+1

If x=5m+2

Then x

2

=(5m+2)

2

=25m

2

+20m+4

=5(5m

2

+4m)+4

=5n+4 [ Taking n=5m

2

+4m]

If x=5m+3

Then x

2

=(5m+3)

2

=25m

2

+30m+9

=5(5m

2

+6m+1)+4

=5n+4 [ Taking n=5m

2

+6m+1]

If x=5m+4

On squaring both side and we get,

x

2

=(5m+4)

2

=25m

2

+16+40m

=5(5m

2

+8m+3)+1(where5m

2

+8m+3=n)

=5n+1

Hence, In each cases x

2

is either of the of the form 5n or 5n+1 for integer n..

thank you….