Question

show that the square of any positive integer is of the form pq +1, pq + 4 for some integer q

Answer 1

Step-by-step explanation:

Let 'a' be any positive integer.

b=4

by Euclid's division lemma,

a=bq+r

a²=(bq+r)²-------1.

r=0,1,2,3

from 1.

for r=0,

a²=(4q+0)²

a²=16q²

a²=4(4q²)

=4q, where q=4q².

for r=1,

a²=(4q+1)²

a²=16q²+1+8q

a²=4(4q²+2q)+1

a²=4q+1, where q=4q²+2q.