Question

show that any positive integer is in the form 3q or 3q 1 or 3q 2 for some integer q

Answer 1

Hiii Friend,

Let n be an arbitrary positive integer.


On dividing n by 3 , let Q be the Quotient and r be the Remainder .

Then by Euclid's division lemma , we have

n = 3Q+r , where r = 0,1,2

When r = 0

N = 3Q

When r = 1

N = 3Q+1

When r = 2

N = 3Q+2


Therefore,

N = 3Q or (3Q+1) or (3Q+2) , for some integer Q.


Thus, any positive integer is in the form of 3Q , (3Q+1) or (3Q+2) for some integer Q.


HOPE IT WILL HELP YOU ..... :-)

Answer 2

To Show :

Any positive integer is of the form 3q or 3q+1 or 3q+2 .

Solution :

Let a be any positive integer .

Then b = 3

So by Euclid's Division lemma there exist integers q and r such that ,

a = bq+r

a = 3q+r (b = 3)

And now ,

As we know that according to Euclid's Division Lemma :

0 ≤ r < b

Here ,

0 ≤ r < 3

Here the possible values of r are = 0,1,2

=> 0 ≤ r < 1<2

=> r = 0 or r = 1 or r = 2

And then

a = 3q+r

a = 3q+0 = 3q

a = 3q+1

a = 3q+2

#Hence Proved !!