Prove that when the square of any natural number leaves the remainder either 0 or 1 when divided by 4

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Prove that when the square of any natural number leaves the remainder either 0 or 1 when divided by 4

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Reagan 5 days 2021-09-14T07:01:53+00:00 1 Answer 0 views 0

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    2021-09-14T07:02:55+00:00

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    Every integer when squared leaves a the remainder 0 or 1 when divided by 4.

    Here, we have two kinds of integers:

    Even integers,

    Odd integers,

    An even integer can be expressed in the form 2n

    And an odd integer can be expressed in the form 2n + 1

    So, square of an even integer is =

    Square of an odd integer is = + 4n + 1

    = 0 mod 4

    4( + n) + 1 = 1 mod 4

    So, every even integer squared leaves a remainder 0 when divided by 4.

    And every odd integer squared leaves a remainder 1 when divided by 4.

    Hence, any integer squared leaves a reminder 0 or 1 when divided by 4.

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