Is it true that the square of an odd positive integer is of the form 8q +1, for some integer q?

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Accepted Answer

Given that a square of an odd positive integer is of the form 8q +1, for some integer q.Euclid's Division Lemma states that if two positive integers a and b exist, then there must be unique values of q and r that satisfy the formula a= bq + r, where  0 ≤ r < b.Let n be a positive integer for some integer q, then n can be written as,n = bq+rFrom Euclid's Division Lemma,b = 4

⇒
  n = 4q +r, where 0 ≤ r ≤ 4                  ……(1) (1) can be of the form 4q or 4q+1.Now, substitute r = 0 in (1),

⇒
  n = 4q, is an even number. Substitute r = 1 in (1),

⇒
  n = 4q+1, is an odd number                                              ……(2) Square both sides in (2),

n
        2

=
        
         (4q+1)
        2

=16
        q
        2
       
       +8q+1

=8
        
         2
          q
          2
         
         +q
       +1

Let (

2
    q
    2

+q) = k, where k is an integer.Then,

n
    2
   
   =8k+1
   Substitute r = 2 in (1),

⇒
  n = 4q+2, which is an even number.  Substitute r = 3 in (1),

⇒
  n = 4q+3,  which is an odd number.                                               ……(3) Square both sides in (3),

n
        2

=
        
         (4q+3)
        2

=16
        q
        2
       
       +24q+9

=8
        
         2
          q
          2
         
         +3q+1
       +1

Let (

2
    q
    2

+3q+1)  =k , where k is an integer.

⇒
    n
    2
   
   =8k+1
   Therefore from (2) and (4), the square of an odd positive integer is of the form 8q + 1, where q is some integer.Therefore, the square of an odd positive integer is of the form 8q + 1, for some integer q is true.Hence the correct option is 1.

Is it true that the square of an odd positive integer is of the form 8q +1, for some integer q?

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