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  • Show that the square of any odd integer is of the form 4q + 1 for some integer q

Show that the square of any odd integer is of the form 4q + 1 for some integer q.

Answers (1)

By Euclid’s division –
Any positive integer can be written as:
A = bq + r  
Here b = 4
r is remainder when we divide A by 4, therefore:
  0 \leq r < 4, r = 0, 1, 2, 3
Even: A = 4q and 4q + 2      
Odd: A = 4q + 1 and 4q + 3              

Case 1:
Now squaring the odd terms, we get
(4q + 1)2 = (4q) 2 + (1)2 + 8q
=   16q2 + 8q + 1
= 4(4q2 + 2q) + 1
= 4m + 1               
Here m = 4q2 + 2q

Case 2:
(4q + 3)2 = (4q)2 + (3)2 +24q
= 16q2 + 8 + 1 + 24q
=16q2 + 24q + 8 + 1
=4(4q2 + 6q + 2) + 1
  =4m + 1                 
Here m = 4q2 + 6q + 2
Hence we can say that square of any odd integer is of the form 4q + 1 for some integer q.

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