Question

Prove that the square of any positive integer is in the form of 3M or 3M + 1 from some integer m

Answer 1

Answer:

Step-by-step explanation:

By Euclid's division lemma, a=bq+r, 0≤r<b

Let,

a=any positive integer

b=3      {0≤r<3}

r=0, 1, 2

Case 1 (r=0):-

a=bq+r

a=3q+0

(a)²=(3q)²

a²=9q²

a²=3(3q²)

a²=3m     {m=3q²}

Case 2 (r=1):-

a=bq+r

a=3q+1

(a)²=(3q+1)²

a²=9q²+6q+1

a²=3(3q²+2q)+1

a²=3m+1    {m=3q²+2q}

Case 3 (r=2):-

a=bq+r

a+3q+2

(a)²=(3q+2)²

a²=9q²+12q+4

a²=3(3q²+4q+1)+1

a²=3m+1      {m=3q²+4q+1}

∴ By Euclid's division lemma, the square of any positive integer is in the form of 3m or 3m+1.