State True/False.Any positive odd integer is of the form 6q+1 or, 6q +3or, 6q +5, where q is some integer.

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Given to check that any positive odd integer is of the form 6q+1 or, 6q +3or, 6q +5, where q is some integer.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+r
   If b = 6

⇒n=6q+r
   , where

0≤r≤6
                     ……(1)Substitute r = 0 in (1),

n=6q

⇒n  is  even  for any positive integer q

⇒n
   is a positive even integer Substitute r = 1 in (1),

n=6q+1
   ……(2)

⇒n  is  odd  for any positive integer q

⇒n
   is a positive odd integerSubstitute r = 2 in (1),

n=6q+2
   …………..(3)

⇒n  is  even  for any positive integer q

⇒n
   is a positive even integer. Substitute r = 3 in (1),

n=6q+3
   ……(4)

⇒n  is  odd  for any positive integer q

⇒n
   is a positive odd integerSubstitute r = 4 in (1),

n=6q+4

⇒n  is  even  for any positive integer q

⇒n
   is a positive even integer.Substitute r = 5 in (1),

n=6q+5
   ……(5)

⇒n  is  odd  for any positive integer q

⇒n
   is a positive odd integerTherefore it is obtained from (2),(4) and (5), for some positive integer q , any positive odd integer is of the form

6q+1
   or

6q+3
   , or,

6q+5
   .Thus, any positive odd integer is of the form

6q+1
   or,

6q+3
   or,

6q+5
   , where q is some integer is true.Hence the correct option is 1.

State True/False.

Any positive odd integer is of the form 6q+1 or, 6q +3or, 6q +5, where q is some integer.

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