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

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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  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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