Question

Show that any positive odd integer is in the form of 8q+1 or 8q+3 or 8q+5 or 8q+7.where q is some integer

Answer 1

Hey there!

let us start with with taking a, where a is a positive odd integer.
We apply the division algorithm with a and b = 8.
since 0 ≤ r < 8 the possible remainders are 0,1,2,....7.
That is a can be 8q or8q+1 or 8q+2 or 8q+3 or 8q+4or 8q+5 or 8q+6 where q is quotient.
How ever since a is odd a cannt be 8q or 8q+2 or 8q+4 (since they are divisible by 2).
There fore, any odd integer is of the form 8q+1, 8q+3, 8q+5 or 8q+7.

Hope it help

Answer 2

Answer:

Step-by-step explanation:

Yes

Let a be any positive integer

According to Euclids division lemma

a = bq + r

b = 8

so the possible values of r = 0,1,2,3,4,5,6,7

a = 8q ( it is even)

a = 8q + 1 ( it is odd)

..........and so on