Question

2.
Use division algorithm to show that any positive odd integer is of the form 6q+1,
or 6q + 3 or 69 +5, where q is some integer.​

Answer 1

Answer:

As per Euclid's division lemma

if a and b are 2 positive integer ,then

a=bq+r

where 0<r<b

let positive integer be a

And b=a

Hence. a=6q+r

where(0<r<6)

r is an integer greater than or equal to 0 less than 6

hence. r can be either 0,1,2,3,4 or 5

Answer 2

Let

′

a

′

be any positive integer and b=6

Then by division algorithm

a=6q+r where r=0,1,2,3,4,5

so, a is of the form 6q or 6q+1 or 6q+2 or 6q+3 or

6q+3 or 6q+4 or 6q+5

Therefore If s is an odd integer

Then

′

a

′

is of the form 6q+1 or 6q+3 6q+5

Hence a positive odd integer is of the form 6q+1 or 6q+3 or 6q+5