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(NCERT)

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

integer.

Question

(NCERT)

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

integer.

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Mathematics
2 months
2021-08-03T13:16:02+00:00
2021-08-03T13:16:02+00:00 2 Answers
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## Answers ( )

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:

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.