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

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1. 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….

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