Question

2. use divison algorithm to show that any positve odd integer
is of the form 6q+1 0r 6q+3 Or 6q+5 where q is some
integer,​

Answer 1

Answer:

let the integer be a

according to euclied's division lemma,

a=bq+r

let b=6

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

therefore the numbers are

6q+1,6q+2,6q+3,6q+4,6q+5

when a=6q,

since q is an integer and 6 is an even number 6q is even

when a=6q+1,

since 6q is even, 6q+1 is odd

when a=6q+2

since 6q is even 6q+2 is even

when a=6q+3,

since 6q is even, 6q+3 is odd

when a=6q+4

since 6q is even 6q+4 is even

when a=6q+5

since 6q is even, 6q+5 is odd

thus proved