Question

Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m. (Please show some calculation)

Answer 1

ANSWER

Let a be the positive integer and b=3.

By using Euclid’s Divison Lemma

We know a=bq+r, 0≤r<b

Now, a=3q+r, 0≤r<3

The possibilities of remainder is 0,1, or 2.

Case 1 : When a=3q +0,

a2=(3q)2 =9q2 =(3m)²where m=3q² :. (Squaring both side)

Case 2 : When a=3q+1

a²=(3q+1)²=(3q)²+(2×3q×1)+(1)²

=9q²+6q+1. :. (Squaring both side)

=3(3q²+2q)+1 where m=q(3q+2) (:. By taking 3 as common factor)

Case 3: When a=3q+2

a²=(3q+2)²=(3q)²+(2*3q*2)+2²

=9q²+12q+3+1

=3(3q²+4q+1)+1. (:. By taking common 3 as factor)

= 3m+1

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Answer 2

Answer:

It is the correct answer.

Step-by-step explanation:

Hope this attachment helps you.