Question

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

Answer 1

As per Euclid's Division Lemma
If a and b are two positive integers, then
a = bq + r
where 0 < r < b

Let positive integer be a
and b = 4
hence, a = 4q + r where ( 0 < r < 4 )
r is an integer greater than or equal to 0 and less than 4
hence r can be either 0, 1, 2 or 3
If r = 1
our equation becomes
a = 4q + r
a = 4q + 1
this will always be an odd integer
when r = 3
our equation becomes
a = 4q + 3
this will always be an odd integer
Therefore any integer is of the form 4q + 1 or 4q + 3
Hence Proved

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 .