Question

Show that the square of an odd positive integer is of the form 8m+1, for some whole number m.​

Answer 1

Step-by-step explanation:

Let a be any positive integer and b = 4.

Then, by Euclid's algorithm a = 4q + r for some integer q  0 and 0  r < 4  

Thus, r = 0, 1, 2, 3  

Since, a is an odd integer, so a = 4q + 1 or 4q + 3

Case I: When a = 4q + 1  

Squaring both sides, we have, a2 = (4q + 1)2  

a2 = 16q2 + 1 + 8q  

   = 8(2q2 + q) + 1  

    = 8m + 1, where m = 2q2 + q  

Case II: When a = 4q + 3

Squaring both sides, we have,  

a2 = (4q +3)2  

   = 16q2 + 9 + 24q  

   = 16 q2 + 24q + 8 + 1  

    = 8(2q2 + 3q + 1) +1  

    = 8m +1 where m = 2q2 + 3q + 1

Hence, a is of the form 8m + 1 for some integer m.