Question

show that any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5 Where q is some integer.

Answer 1

Hlo mate :-

Solution :-
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● Let a be the positive odd integer when divided by 6 gives "q" as quotient and "r"as remainder.

a/c 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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Answer 2

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$\huge{ \mathfrak{ \blue{here \: is \: answer}}}$

let a be any positive integer

then

b= 6

a= bq+r

0≤r<b

0≤r<6

r= 0,1,2,3,4,5

case 1.

r=0

a= bq+r

6q+0

6q

case 2.

r=1

a= 6q+1

6q+1

case3.

r=2

a=6q+2

case 4.

r=3

a=6q+3

case 5

r=4

a=6q+4

case 6..

r=5

a=6q+5

hence from above it is proved that any positive integer is of the form 6q, 6q+1,6q+2,6q+3,6q+4 and 6q+5and

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