show that any positive odd integer is of form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.​

1 thought on “show that any positive odd integer is of form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.​”

1. Let a be the positive odd integer which when divided by 6 gives q as quotient and r as remainder.

   According to Euclid’s division lemma

   [tex]a = bq + r[/tex]

   [tex]b = 6[/tex]

   Where,

   [tex](0 ≤ r < 6)[/tex]

   So,

   [tex]r = 0,1,2,3,4,5[/tex]

   Case 1:

   [tex]If \: r = 1, \: then [/tex]

   [tex]a = 6q + 1[/tex]

   The Above equation will be always as an odd integer.

   Case 2:

   [tex]If \: r=3, \: then[/tex]

   [tex]a = 6q + 3[/tex]

   The Above equation will be always as an odd integer.

   Case 3:

   [tex]If \: r=5, \: then[/tex]

   [tex]a = 6q + 5[/tex]

   The Above equation will be always as an odd integer.

   ∴ Any odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5.

   Hence proved.

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