Question

Show that every positive odd positive integer can be written in form of 4q+1 or 4q+3

Answer 1

let "a " be any positive odd integers and b=4 by using Euclid;,s division algorithm a=bq+r o<r <4 ☆the possible value of r= 0,1,2,3 in which a= 4q,4q+1,4q+2,4q+3 in which 4q+1,4q+3 is a positive odd integers

Answer 2

Step-by-step explanation:

Let a be the positive integer.

And, b = 4 .

Then by Euclid's division lemma,

We can write a = 4q + r ,for some integer q and 0 ≤ r < 4 .

°•° Then, possible values of r is 0, 1, 2, 3 .

Taking r = 0 .

a = 4q .

Taking r = 1 .

a = 4q + 1 .

Taking r = 2

a = 4q + 2 .

Taking r = 3 .

a = 4q + 3 .

But a is an odd positive integer, so a can't be 4q , or 4q + 2 [ As these are even ] .

•°• a can be of the form 4q + 1 or 4q + 3 for some integer q .

Hence , it is solved .