Question

show that the square of an odd positive is of the form 18 + 1 for some integer q​

Answer 1

Answer:

Step-by-step explanation:

By Euclids division lemma, given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

Take ,  

b=8

a=8q+r

Hence  

r=0,1,2.....7

Case(i): If  

r=0

a=8q

⇒a²=64q²

=8(8q²)

= 8m where m= 8q²

 

Case (ii): If  

r=1

a=8q+1

⇒a²=(8q+1)²=64q²+16q+1

=8(8q²+2q)+1)

=8m+1 where m = 8q²+2q