Question

show that any positive odd number is of the form 6q+1, 6q+1, 6q+5. where q is some integer

Answer 1

hey dear


here is your answer


Let a is positive odd integer which when divided by 6 gives q and quotient and r as remainder



using Euclid division lemma

a = bq + r


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

then

a= 6q

a = 6q +1


a = 6q +2

a= 6q + 3

a = 6q + 4

a = 6q +5


here we see that ( 6q + 1 , 6q + 3 , 6q +5 )

these are odd integer


hence shown

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hope it helps

thank you

Answer 2

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.