Show that any positive odd integer is of the form 6q+1, or 6q + 3, or 6q+5, where q is some integer.
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2 thoughts on “Show that any positive odd integer is of the form 6q+1, or 6q + 3, or 6q+5, where q is some integer.<br />​”

1. Answer:

   Let a be a given integer.

   On dividing a by 6 , we get q as the quotient and r as the remainder such that

   a = 6q + r, r = 0,1,2,3,4,5

   when r=0

   a = 6q,even no

   when r=1

   a = 6q + 1, odd no

   when r=2

   a = 6q + 2, even no

   when r = 3

   a=6q + 3,odd no

   when r=4

   a=6q + 4,even no

   when r=5,

   a= 6q + 5 , odd no

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

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2. [tex]\huge\mathcal\colorbox{pink}{{\color{b}{✿Yøur-AñsWēR♡}}}[/tex]

   According to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique

   integers q and r which satisfies the condition

   a = bq + r where 0 ≤ r < b.

   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

   a = bq + r

   a = 6q + r………………….(1)

   where, (0 ≤ r < 6)

   So r can be either 0, 1, 2, 3, 4 and 5.

   Case 1:

   If r = 1, then equation (1) becomes

   a = 6q + 1

   The Above equation will be always as an odd integer.

   Case 2:

   If r = 3, then equation (1) becomes

   a = 6q + 3

   The Above equation will be always as an odd integer.

   Case 3:

   If r = 5, then equation (1) becomes

   a = 6q + 5

   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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