(NCERT)
4. Show that any positive odd integer is of the form 6q +1,6q+ 3 or 6q+5, where q is some
integer.​

2 thoughts on “(NCERT)<br />4. Show that any positive odd integer is of the form 6q +1,6q+ 3 or 6q+5, where q is some<br />integer.​”

1. Answer:

   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.

   Step-by-step explanation:

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2. Step-by-step explanation:

   We know that,

   a = bq+r. {by Euclid’s division lemma}

   let the first integer be 6q,

   second be 6q+1

   third be 6q+2

   fourth be 6q+3

   fifth be 6q+4

   sixth be 6q+5

   From above we can see that second, fourth, and sixth are odd positive integers and can be written in the form 6q +1,6q+ 3 or 6q+5, while first, third,fifth cannot.

   Hence proved.

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