Question

Use Euclid division lemma to show that the square of any positive integer is either of the form 3m or 3m +1 for some integer m

Sahi uttar dedo

plz ​

Answer 1

Answer:

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Let a be the positive integer and b=3.

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

a²=(3q)² =9q² =3q×3q=3m where m=3q²

Case 2 : When a=3q+1

a²=(3q+1)²=(3q)²+(2×3q×1)+(1)²=3q(3q+2)+1=3m+1 where m=q(3q+2)

Case 3: When a=3q+2

a²=(3q+2)²=(3q)²+(2×3q×2)+(2) ²=9q² +12q+4=9q²+12q+3+1=3(3q²+4q+1)+1=3m+1

where m=3q²+4q+1

Hence, from all the above cases, it is clear that square of any positive integer is of the form 3m or 3m+1.