Question

Show that any positive odd integers is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer

Answer 1

let a be any positive integer and b= 6. By Euclid's Division Lemma a=bq+r. When r=0, then a= 6q+0

a= 6q. when r= 1, a= 6q+1 when r= 2, a= 6q+2. when r= 3. a= 6q+3

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.