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. About the author Remi

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…. Reply

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….