Question

Use euclid's lemma to show that the square of any positive integer is in the form of 4 m or 4 m + 1 for some integer m

Answer 1

Step-by-step explanation:

Let "A" be any positive integer and "b=4".

Apply edl for this.

Possible remainders are 0,1,2 or 3.

Therefore A can be in the form of

4q+1,4q+2,4q+3

Now, (4q)2=4(4)*Q2

=16q2

=4(4q2)

=4m

Again,(4q+1)2=(4q)2+1+8q

=16q2+8q+1

=4(4q2+2q)+1

=4m+1

Therefore square of all positive integers is of the form 4m,4m+1.

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