Question

Show that any positive even integer is of the form 4 q or 40 + 2 and any positive odd number is of the form 4 q

Answer 1

Let a be any positive integer
Applying E.D.L we get
a=bq+r..........
For b=4
therefore. a=4q+r.............. (i) [0<=r<4]
when r=0 then
a=4q............even integer
when r=1
a=4q+1........ odd integer
when r=2
a=4q+2........even integer
when r=3
a=4q+3.........odd integer
therefore it is clearly seen in the above that
4q or 4q+2 is in the form of even integer
and 4q+1 or 4q+3 is in the form of odd integer
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 even positive integer, so a can't be 4q + 1 , or 4q + 3 [ As these are odd ] .

∴ a can be of the form 4q or 4q + 2 for some integer q .

Hence , it is solved