Question

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

Answer 1

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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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