Question

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Show That Any Positive Odd Integers Is Of The Form 6q+1, or 6q+3, or 6q+5, Where q is Some Integers.?


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Answer 1

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.

Answer 2

We Consider An positive integer as a.

On dividing a by b .

Here ,

let q is the quotient and r is the remainder.

Now,

a = bq + r , 0 ≤ r < b ..... ( 1 )

Here we putting b = 6 in eq ( 1 )

Here we find ,

a = 6q + r , 0 ≤ r < b ..... ( 2 )

so here possible values of r = 1 , 2, 3, 4, 5.

If r = 0, then find Equation (2) , a = 6q.

Here,

6q is is divisible by 2 , so 6q is here Even .

If r = 1 , then find Equation (2) , a = 6q + 1.

Here,

6q + 1 is not divisible by 2 , so 6q + 1 is here odd.

If r = 2 , then find Equation (2) , a = 6q + 2.

Here,

6q + 2 is not divisible by 2 , so 6q + 2 is here even.

If r = 3 , then find Equation (2) , a = 6q + 3.

Here,

6q + 3 is not divisible by 2 , so 6q + 3 is here odd.

If r = 4 , then find Equation (2) , a = 6q + 4.

Here,

6q + 4 is not divisible by 2 , so 6q + 4 is here even.

If r = 5 , then find Equation (2) , a = 6q + 5.

Here,

6q + 5 is not divisible by 2 , so 6q + 5 is here odd.

so , a is odd , so a cannot be 6q,6q+2, 6q+4.

Therefore any positive odd integer is of the form 6q+1 , 6q +3, 6q + 5.

Here we proved.

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