Question

show that square of any positive integer is of the form 3m or 3m + 1 for some integer​

Answer 1

Step-by-step explanation:

If a and b are two positive integers, then,

a = bq +r, 0 < r

<b Let b = 3

Therefore, r = 0, 1,2

Therefore, a =

3q or a = 3q + 1 or a = 3q + 2

If a = 3q

a2 = 9q2 = 3(3q2) = 3m where m =

If a = 3q + 1 a2 = 9q2 + 69 + 1 = 3(3q2 + 2q) + 1 =

3m + 1 where m = 3q2 + 2q

If a = 3q + 2 a? = 9q? + 12q + 4 = 3(3q2 + 4q + 1) +

1= 3m + 1, where m = 3q2 + 4q + 1

Therefore, the square of any positive integer is either

of the form 3m or 3m + 1.