Question

Show that any positive odd integer is of the form 4q+1 or 4q+3 where q is some integer.........



Best answer only

Answer 1

let a be the positive odd integer and b=4
0_<r<4
r=0,1,2,3
I.e.for r=o,a=4q
for r=1,a=4q+1
for r=2,a=4q+2
for r=3,a=4q+3
a is odd , a cannot be 4q or 4 q+ 2( because they are both divisible by 2).
therefore any odd integer is of the form 4q+1 or 4 + 3.

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 .