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

Step-by-step explanation:

Let a be any positive integer and b = 6

Then we use here Euclids Lemma

a=bq+r, q>0 and 0<r<6

therefore r can be =0,1,2,3,4,5

if r=0 then, a=6q+0=6q=2*3q is even

if r=1 then, a=6q+1=(2*3q)+1 is odd

if r=2 then, a=6q+2 =6q+1+1 is ((2*3q)+1)+1 is even

if r =3 then, a=6q+2+1=((2*3q)+2)+1 is odd

if r=4 then, a=6q+4=2*(3q+2) is even

if r=5 then, a=6q+5=(6q+4)+1 is odd

Hence odd entries are of the form =

(6q+1) ,(6q+3) ,(6q+5)

Hope u will learn how to solve this types of questions

Answer 2

$\huge \underline \mathbb {SOLUTION:-}$

Let a be any positive integer and b = 6. Then, by Euclid’s algorithm, a = 6q + r, for some integer q ≥ 0, and r = 0, 1, 2, 3, 4, 5, because 0≤r<6.

Now substituting the value of r, we get,

If r = 0, then a = 6q

Similarly, for r= 1, 2, 3, 4 and 5, the value of a is 6q+1, 6q+2, 6q+3, 6q+4 and 6q+5, respectively.

If a = 6q, 6q+2, 6q+4, then a is an even number and divisible by 2. A positive integer can be either even or odd Therefore, any positive odd integer is of the form of 6q+1, 6q+3 and 6q+5, where q is some integer.