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

Answer 1

let ' a' be any positive integer and b = 3.

we know, a = bq + r , 0 < r< b.

now, a = 3q + r , 0<r < 3.

the possibilities of remainder = 0,1 or 2

Case I - a = 3q

a2 = 9q2

= 3 x ( 3q2)

= 3m (where m = 3q2)

Case II - a = 3q +1

a2 = ( 3q +1 )2

Answer 2

Answer:

Step-by-step explanation:

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refer attachment