2. Show that any positive odd integer is of the form 6q+1, or 6q+3, or 69 + 5, where q is
some integer
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2 thoughts on “2. Show that any positive odd integer is of the form 6q+1, or 6q+3, or 69 + 5, where q is<br />some integer<br />​”

1. Step-by-step explanation:

   since 6 is a even number when we multiply it with any number we get the answer as a even number and when we add a odd number to a even number we will all ways get odd number.

   Hence proved

   Hope you understand

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2. Answer:

   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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