Question

Show that any positive integer can be written as in the form of 6m+1 ,6m+3or 6m +5 ,where m is an positive integer?

Answer 1

Let a be any of positive integer and b=6.then there exist integers q and r such that

a=6q+r,0_<6

a=6q or 6q+1 or 6q+2 or 6q+3 or 6q+4 or 6q+5

But 6q,6q+2 and 6q+4 are even positive integers so

a=6q+1 or 6q+3 or 6q+5

Answer 2

Answer:

Let a be any positive integer and b=6

Now by Euclid Division Lemma we know that,

a= bq+r & 0<=r<b

Case 1- when r =0

Then, a= 6m

Case 2- when r = 1

Then, a= 6m+1

Case 3 - when r = 2

Then, a= 6m+2

Case 4- when r = 3

Then, a= 6m+3

Case 5- when r = 4

Then, a= 6m+4

Case 6 - when r = 5

Then, a= 6m+5

So, we see that every positive integer can be written in the form of 6m,6m+1,6m+3 & 6m+5.