Question

8. Prove that the square of any positive integer is of the form 4q or 4q+1 for some integer q.​

Answer 1

Answer:

Let positive integer a be the any positive integer. Then, b = 4 . 0 ≤ r < 4 , So r = 0, 1, 2, 3. Hence ,Square of any positive integer is in form of 4q or 4q + 1 , where q is any integer.

Step-by-step explanation:

Let 'a' be any positive integer.

b=4

by Euclid's division lemma,

a=bq+r

a²=(bq+r)---1.

r=0,1,2,3 from

1. for r=0,

a-(4q+0)?

a²=16q?

a²=4(4q) =4q,

where q=4q?.

for r=1,

a-(4q+1)?

a-16q?+1+8q a?

-4(4q+2q)+1 a?=4q+1,

where q=4q²+2q.