Question

show that any positive odd integer is of the form 5q + 1, 5q + 3 where Q is some integer​

Answer 1

Step-by-step explanation:

let 'a' be any positive odd integer & a = 5

Then by Euclid's Division Algorithm,

a = bq + r ,

( 0 less than or equal to r less than b)

a = 5q + r

( 0 less than or equal to r less than 5)

So, r = 0,1,2,3,4

a = 5q + 0

a = 5q

Here, 5q is exactly divisible by 2 , hence , it is an even positive number.

a = 5q + 1

Here, 5q + 1 is not exactly divisible by 2 , hence , it is an odd positive number.

a = 5q + 2

Here, 5q + 2 is exactly divisible by 2 , hence , it is an even positive number.

a = 5q + 3

Here, 5q + 3 is not exactly divisible by 2 , hence , it is an odd positive number.

The procedure continues. Hence , any positive integer will be of form 5q + 1, 5q + 3... where q is some integer.

Hope it helps!