Question

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

Answer 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
a=bq+r

a=6q+r

where , a=0,1,2,3,4,5
then,
a=6q
or
a=6q+1
or
a=6q+2
or
a=6q+3
or
a=6q+4
or
a=6q+5

but here,
a=6q+1 & a=6q+3 & a=6q+5 are odd.

Answer 2

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QUESTION:show that any positive odd integer is of the form 6q+1,6q+3 or 6q+5 for some integer q

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ANSWER:

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

where , a=0,1,2,3,4,5

then,

a=6q

or

a=6q+1

or

a=6q+2

or

a=6q+3

or

a=6q+4

or

a=6q+5

but here,

a=6q+1 & a=6q+3 & a=6q+5 are odd.

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